arX

iv0

705

2366

v1 [

hep-

th]

16

May

200

7

MPP-2007-57LMU-ASC 3107

Instantons and Holomorphic Couplings in

Intersecting D-brane Models

Nikolas Akerblom1 Ralph Blumenhagen1 Dieter Lust12Maximilian Schmidt-Sommerfeld1

1 Max-Planck-Institut fur Physik Fohringer Ring 6

80805 Munchen Germany

2 Arnold-Sommerfeld-Center for Theoretical Physics

Department fur Physik Ludwig-Maximilians-Universitat Munchen

Theresienstraszlige 37 80333 Munchen Germany

akerblom blumenha luest pumucklmppmumpgde

Abstract

We clarify certain aspects and discuss extensions of the recently introducedstring D-instanton calculus (hep-th0609191) The one-loop determinantsare related to one-loop open string threshold corrections in intersecting D6-brane models Utilising a non-renormalisation theorem for the holomorphicWilsonian gauge kinetic functions we derive a number of constraints forthe moduli dependence of the matter field Kahler potentials of intersectingD6-brane models on the torus Moreover we compute string one-loopcorrections to the Fayet-Iliopoulos terms on the D6-branes finding thatthey are proportional to the gauge threshold corrections Employing theseresults we discuss the issue of holomorphy for E2-instanton corrections tothe superpotential Eventually we discuss E2-instanton corrections to thegauge kinetic functions and the FI-terms

Contents

1 Introduction 2

2 A non-renormalisation theorem 4

3 One-loop thresholds for intersecting D6-branes on T6 6

4 Fayet-Iliopoulos terms 8

5 Wilsonian gauge kinetic function and σ-model anomalies 9

51 Holomorphic gauge couplings for toroidal models 1152 Universal threshold corrections 13

6 Holomorphic E2-instanton amplitudes 16

61 Half-BPS instantons 1662 Superpotential contributions 1763 Instanton corrections to the gauge kinetic functions 2064 Instanton corrections to the FI-terms 22

7 Conclusions 23

1 Introduction

Type IIA orientifolds with intersecting D6-branes and their mirror symmetricType IIB counterparts have proven to provide a phenomenologically interestingclass of string compactifications and have been under intense investigation duringthe last couple of years [1 2 3 4]

In order to make contact with experiment one needs not only the means todetermine the gauge group and chiral matter content of such a string model butalso has to develop tools for the computation of the low energy effective actionIt is in this latter low energy description where important issues like modulistabilisation and supersymmetry breaking are discussed

In this paper we would like to clarify certain important aspects of this effectiveaction which to our knowledge have so far not been spelled out in the literatureThe first issue concerns the properties of the gauge couplings in N = 1 supersym-metric D6-brane vacua The physical one-loop open string threshold correctionsto the gauge couplings have been computed in [5 6] for toroidal backgroundsHere we first state a non-renormalisation theorem for the holomorphic gauge ki-netic function at the one-loop level and then explicitly show that it is indeedsatisfied for the Wilsonian gauge couplings in toroidal Type IIA orientifolds

We revisit perturbative one-loop corrections to the Fayet-Iliopoulos (FI) termsfor intersecting D6-branes In [7] it was shown that if the D6-branes are su-

2

persymmetric at tree-level in a globally consistent model no such correctionsare generated at one-loop We ask the question whether a small non-vanishingFI-term on a D6b-brane can induce an FI-term on a D6a-brane which is super-symmetric at tree-level We find the intriguing result that the one-loop inducedFI-term on brane D6a can be expressed by the gauge threshold corrections

Moving back to gauge couplings the extraction of the Wilsonian part involvesan interesting interplay between the non-holomorphic gauge couplings and theKahler potentials for all the matter fields involved providing strong constraintson the complete moduli dependence of the matter field Kahler potentials As wewill see in order to cancel all σ-model anomalies in the effective action a one-loop redefinition of the dilaton S-field as well as of the complex structure moduliU is needed The hereby induced corrections to the gauge coupling constants willbe referred to as ldquouniversalrdquo threshold corrections (in analogy with the heteroticstring) This is in contrast to perturbative heterotic string compactificationswhere only the dilaton acquires a universal one-loop field redefinition [8 9 1011 12 13]

Having revisited and discussed the perturbative one-loop corrections in thesecond part of the paper we undertake some first steps towards a better un-derstanding of possible D-instanton effects Such effects are very important foran understanding of the vacuum structure (these days called the landscape) ofstring compactifications and it has been pointed out recently that they can alsogenerate phenomenologically appealing terms like Majorana masses for neutrinos[14 15 16 17 18 19 20] Moreover they are also important for the string the-ory description of gauge instanton effects [21 22 23 24 25] (see [26] for a recentgeneral review on instantons)

String instantons are given by wrapped string world-sheets as well as by wrap-ped Euclidean D-branes and like in field theory their contributions to the space-time superpotential are quite restricted These contributions can be computed ina semi-classical approach ie one involving only the tree-level instanton actionand a one-loop determinant for the fluctuations around the instanton [27] Fortype IIA orientifold models on CalabindashYau spaces with intersecting D6-branes(and their T-dual cousins) the contribution of wrapped Euclidean D2-braneshereafter called E2-branes to the superpotential has been determined in [14] (seealso [15]) Since both the D6-branes and the E2-instantons are described by openstring theories it was shown that (in the spirit of the D(minus1) instantons treatedeg in [28 29 21]) the entire instanton computation boils down to the evaluationof disc and one-loop string diagrams with boundary (changing) operators insertedHere both the D6-branes and the E2-instantons wrap compact three-cycles of theCalabindashYau manifold

Intriguingly the one-loop contributions in the instanton amplitude [14] havebeen shown to be identical to string threshold corrections for the gauge couplingsof the corresponding D6-branes [17 23] This relates the computation of suchinstanton amplitudes to the discussion in the first half of this paper So far it has

3

not been explained explicitly in which sense the computed instantonic correlationfunctions are meant to be holomorphic With the results from the first part ofthis paper we clarify this point

Finally we show that E2-instantons not only contribute to the superpotentialbut from the zero mode counting can also contribute to the holomorphic gaugekinetic functions for the SU(N) gauge groups localised on the D6-branes Inorder for such corrections to arise the E2-instanton must not be rigid but mustadmit one extra pair of fermionic zero modes arising from a deformation of theinstanton This is the space-time instanton generalisation of a fact known fromtopological string theory namely that world-sheet instantons induce tr(W 2)hminus1

couplings if they have h boundaries We will see that such couplings can also arisefrom space-time E2-instantons Finally we find that the zero mode counting alsoallows E2-instanton corrections to the FI-terms on the D6-branes Similar to theone-loop corrections these can arise once the supersymmetry on the E2-braneis softly broken by for instance turning on the C3-form modulus through theworld-volume of E2

2 A non-renormalisation theorem

Let us investigate the structure of perturbative and non-perturbative correctionsto the holomorphic gauge kinetic functions for Type II orientifolds We discussthis for Type IIA orientifolds but this is of course related via mirror symmetryto the corresponding Type IIB orientifolds

Consider a Type IIA orientifold with O6-planes and intersecting D6-branespreserving N = 1 supersymmetry in four dimensions ie the D6-branes wrapspecial Lagrangian (sLag) three-cycles Πa of the underlying CalabindashYau manifoldX all preserving the same supersymmetry On the threefold we introduce in theusual way a symplectic basis (AI B

I) I = 0 1 h21 of homological three-cycles with the topological intersection numbers

AI BJ = δJI (21)

Moreover we assume that the AI cycles are invariant under the orientifold pro-jection and that the BJ cycles are projected out The complexified complexstructure moduli on such an orientifold are defined as

U cI =

1

(2π) ℓ3s

[eminusφ4

int

AI

real(Ω3) minus i

int

AI

C3

] (22)

where Ω3 denotes the normalised holomorphic three-form on X and the four-dimensional dilaton is defined by φ4 = φ10 minus 1

2ln(VXℓ

6s) Expanding a three

cycle Πa into the symplectic basis

Πa =M Ia AI +NaI B

I (23)

4

with M I NI isin Z from dimensional reduction of the Dirac-Born-Infeld (DBI)action one can deduce the SU(Na) gauge kinetic functions at string tree-level

fa =

h21sum

I=0

M Ia U

cI (24)

Since the imaginary parts of the U cI are axionic fields they enjoy a Peccei-Quinn

shift symmetry U cI rarr U c

I + cI which is preserved perturbatively and only brokenby E2-brane instantons

Let Ci denote a basis of anti-invariant 2-cycles ie Ci isin H11minus The complex-

ified Kahler moduli are then defined as

T ci =

1

ℓ2s

(int

Ci

J2 minus i

int

Ci

B2

) (25)

where B2 denotes the NS-NS two-form of the Type IIA superstring Thereforealso the complexified Kahler moduli enjoy a Peccei-Quinn shift symmetry brokenby world-sheet instantons Note that the chiral fields T c

i organise the σ-modelperturbation theory and do not contain the dilaton so that the string pertur-bative theory is entirely defined by powers of the U c

I Moreover to shorten thenotation we denote by U c

I and T ci the complexified moduli and by UI and Ti only

the real partsThe superpotentialW and the gauge kinetic function f in the four-dimensional

effective supergravity action are holomorphic quantities In the usual way em-ploying holomorphy and the Peccei-Quinn symmetries above one arrives at thefollowing two non-renormalisation theorems

The superpotential can only have the following dependence on U cI and T c

i

W = Wtree +W np(eminusUc

I eminusT ci

) (26)

ie beyond tree-level there can only be non-perturbative contributions fromworld-sheet and E2-brane instantons Similarly the holomorphic gauge kineticfunction must look like

fa =sum

I

M IaU

cI + f 1-loop

a

(eminusT c

i

)+ fnp

a

(eminusUc

I eminusT ci

) (27)

ie in particular its one-loop correction must not depend on the complex structuremoduli Finally we consider the Fayet-Iliopoulos terms for the U(1)a gaugefields on the D6-branes At string tree-level and for small deviations from thesupersymmetry locus these are given by

ξa = eminusφ4

int

Πa

image(Ω3) = eminusφ4 N Ia

int

BI

image(Ω3) (28)

5

and therefore only depend on the complex structure moduli At this classicallevel there are no αprime corrections It is an important question about brane stabilitywhether these FI-terms receive perturbative or non-perturbative corrections in gsAgain non-renormalisation theorems say that in the Wilsonian sense one expectsperturbative corrections at most at one-loop

In the following we will be concerned with the terms beyond tree-level appear-ing in (26) (27) and for the FI-terms First we discuss the one-loop thresholdcorrections f 1minusloop

(eminusT c

i

) which also make their appearance in the space-time

instanton generated superpotential W np(eminusUc

I eminusT ci

) Second we will revisit the

computation of stringy one-loop corrections to the FI-terms Finally we willdiscuss fnp

(eminusUc

I eminusT ci

)as well as instanton corrections to the FI-terms

3 One-loop thresholds for intersecting D6-branes

on T6

The purpose of this section is to recall the one-loop results for the gauge thresh-old corrections in intersecting D6-brane models [5 30 6] The gauge couplingconstants of the various gauge group factors Ga in such a model up to one loophave the form

8π2

g2a(micro)=

8π2

g2astring+ba2

ln

(M2

s

micro2

)+

∆a

2 (31)

where ba is the beta function coefficient The first term corresponds to the gaugecoupling constant at the string scale which contains the tree-level gauge couplingas well as the ldquouniversalrdquo contributions at one-loop (see section 52) Thesecontributions are universal in the sense that they originate from a redefinition ofthe dilaton and complex structure moduli at one-loop The redefinition is branestack and therefore gauge group independent However as the gauge couplingsdiffer for the various gauge groups already at tree level this correction effectivelyis gauge group dependent The second term gives the usual one-loop running ofthe coupling constants and the third term denotes the one-loop string thresholdcorrections originating from integrating out massive string excitations The lasttwo terms can be computed as a sum of all annulus and Mobius diagrams withone boundary on brane a in the presence of a background magnetic field in thefour-dimensional space-time

ba ln

(M2

s

micro2

)+∆a =

sum

b

TA(D6aD6b) +sum

bprime

TA(D6aD6bprime) (32)

+ TA(D6aD6aprime) + TM(D6aO6)

Here D6cprime denotes the orientifold image of brane c In an orbifold one also hasto take into account the orbifold images of the branes and orientifold planes

6

The relevant amplitudes for the Z2timesZ2 orbifold have been computed [5 6] Ina sector preserving N = 1 supersymmetry (this means in particular

sumI θ

Iab = 0)

the annulus and Mobius amplitudes are (after subtracting terms which uponsumming over all diagrams vanish due to the tadpole cancellation condition) [6]

TA(D6aD6b) =IabNb

2

[ln

(M2

s

micro2

) 3sum

I=1

sign(θIab)minus

minus ln3prod

I=1

(Γ(|θIab|)

Γ(1minus |θIab|)

)sign(θIab)

minus3sum

I=1

sign(θIab) (ln 2minus γ)

] (33)

TM(D6aO6k) = plusmnIaO6k

[ln

(M2

s

micro2

) 3sum

I=1

sign(θIaO6k)minus

minus ln3prod

I=1

(Γ(2|θIaO6k

|)Γ(1minus 2|θIaO6k

|)

)sign(θIaO6k

)

+3sum

I=1

sign(θIaO6k)(γ minus 3 ln 2)

] (34)

where Iab is the intersection number of branes a and b Nb is the number ofbranes on stack b and πθIab is the intersection angle of branes a and b on the Irsquothtorus Similarly IaO6k denotes the intersection number of brane a and orientifoldplane k and πθIaO6k

their intersection angle The formula for TM is only valid for|θIaO6k

| lt 12 the formulas for other cases look similar [6]In a sector preserving N = 2 supersymmetry one finds [5]

TA(D6aD6b) = Nb|IJab IKab |[ln

(M2

s

micro2

)minus ln |η(i T c

I )|4 minus ln(TI VaI ) + γE minus ln(4π)

]

(35)where I denotes the torus on which the branes lie on top of each other TI itsKahler modulus and T c

I its complexification with TI = real(T cI ) Furthermore

V aI = |nI

a + iuImIa|2uI with uI the complex structure modulus of the torus and

nIam

Ia the wrapping numbers on the Irsquoth torus Note that the moduli dependence

of the one-loop threshold function in the N = 2 sectors is in complete agreementwith the non-renormalisation theorems of section two (see eq (27)) since theholomorphic part of ∆N=2

a is proportional only to ln η(i T cI )

For theN = 1 sectors the one-loop thresholds in a given open string D6-branesector have the following form (specialising to the case θ12ab gt 0 θ3ab lt 0)

∆a = minus ba16π2

ln

[Γ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab)

] (36)

This expression is a non-holomorphic function of the complex structure moduliU cI Hence for the N = 1 sectors the holomorphic one-loop gauge kinetic func-

tion f 1minusloopa

(eminusT c

i

)vanishes The emergence of the non-holomorphic terms in the

one-loop threshold corrections will be further discussed in section 5

7

4 Fayet-Iliopoulos terms

In this section we investigate one-loop corrections to the FI-terms for a U(1)agauge field on the D6a-brane induced by the presence of other branes D6b Suchcorrections for Type I string vacua have already been studied in [31 32] and thecase of intersecting D6-branes has been discussed in [7] Here we are followingessentially the computational technique of [7] The crucial observation is thatthe vertex operator for the auxiliary D-field in the (0)-ghost picture is simply

given by the internal world-sheet U(1) current ie V(0)Da

= JU(1) Therefore the

one-point function of V(0)Da

on the annulus with boundaries a and b can be writtenas

〈VDa〉 = minus i

2πpartν

int infin

0

dt Zab(ν it)|ν=0 (41)

where Zab(ν it) denotes the annulus partition function with insertion of exp(2πiJ0)in the open string sector (ab) where J0 is the zero mode of the U(1) current Inthe case of intersecting D6-branes on a torus preserving N = 1 supersymmetryand after application of the Riemann theta-identities this partition function isgiven by

Zab(ν it) = IabNb(minusi)3π4t2

ϑ1(3ν2 it)

prodI ϑ1(minusν

2+ iθI

2t it)

η3(it)prod

I ϑ1(iθI2t it)

(42)

Using that ϑ1(0 it) = 0 and ϑprime1(ν it)|ν=0 = minus2πη3 one obtains the divergence

〈VDa〉 ≃ IabNb

int infin

0

dt

t2 (43)

which is cancelled by tadpole cancellation in a global model Therefore oncethe D6-branes are supersymmetric at tree-level no FI-term is generated at one-loop level and the system is not destabilised This result is consistent with thecomputations in [31 32]

However this is not the end of the story of the one-loop corrections to FI-terms in intersecting D6-brane models One can also envision that a tree-levelFI-term on a brane D6b induces via a one-loop diagram an FI-term on a braneD6a To proceed we assume that θIb rarr θIb + 2ǫI with

sumǫI = ǫ and compute

〈VDa〉ǫI = minus i

2πpartνpartǫI

int infin

0

dt Zab(ǫI ν it)|ν=0ǫI=0 (44)

with

Zab(ǫI ν it) = IabNb

(minusi)3π4t2

ϑ1(3ν2+ i ǫ

2t it)

prodI ϑ1(minusν

2+ i (ǫ

IminusǫJminusǫK)2

t+ iθI2t it)

η3(it)prod

I ϑ1(iǫIt+ iθI

2t it)

8

The derivative with respect to the supersymmetry breaking parameters ǫI bringsdown one factor of t and it turns out that the result is the same for all partǫI

〈VDa〉ǫ ≃ i IabNb

int infin

0

dt

t

3sum

I=1

ϑprime1ϑ1

(iθI

abt

2 it2

) (45)

This is the same expression as the one-loop threshold corrections TA(D6a D6b)so that at linear order in ǫ we obtain for the FI-terms

(ξa)ǫ = (ǫb minus ǫa) TA(D6a D6b) (46)

Completely analogously one can show that this formula remains true also forN = 2 open string sectors Therefore we would like to propose that such arelation between gauge threshold and one-loop corrections to FI-terms is validfor general intersecting D6-brane models Moreover the Wilsonian part of thethresholds TA(D6a D6b) which we compute in the next section should also bethe Wilsonian part of the correction to the FI-terms

5 Wilsonian gauge kinetic function and σ-model

anomalies

In a supersymmetric gauge theory one can compute the running gauge couplingsga(micro

2) in terms of the gauge kinetic functions fa the Kahler potential K and theKahler metrics of the charged matter fields Kab(micro2) [33 8 9 10 11 12 13]

8π2

g2a(micro2)

= 8π2real(fa) +ba2

ln

(Λ2

micro2

)+ca2K

+T (Ga) ln gminus2a (micro2)minus

sum

r

Ta(r) ln detKr(micro2) (51)

with

ba =sum

r

nrTa(r)minus 3 T (Ga) ca =sum

r

nrTa(r)minus T (Ga) (52)

and Ta(r) = Tr(T 2(a)) (T(a) being the generators of the gauge group Ga) In addi-

tion T (Ga) = Ta(adjGa) and nr is the number of multiplets in the representationr of the gauge group and the sums run over these representations In this contextthe natural cutoff scale for a field theory is the Planck scale ie Λ2 =M2

PlThe left hand side of eq (51) is given by eq (31) which contains the gauge

coupling at the string scale 1g2astring as well as the one-loop string thresholdcorrections ∆a In general ∆a is the sum of a non-holomorphic term plus thereal-part of a holomorphic threshold correction

∆a = ∆nha + real(∆hol

a ) (53)

9

On the right hand side of eq (51) fa denotes the Wilsonian ie holomorphicgauge kinetic function which is given in terms of a holomorphic tree-level functionplus the holomorphic part of the one-loop threshold corrections (cf eq (27))

fa = f treea + f 1minusloop

a

(eminusT c

i

)=sum

I

M IaU

cI +∆hol

a (54)

In addition on the right hand side of eq (51) the non-holomorphic terms propor-tional to the Kahler metric of the moduli K and the matter field Kahler metricsKr are due to the one-loop contributions of massless fields These fields generatenon-local terms in the one-loop effective action which correspond to one-loopnon-invariances under σ-model transformations the so-called σ-model anomalies(Kahler and reparametrisation anomalies)

Matching up all terms in eq (51) essentially means that the σ-model anoma-lies can be cancelled in a two-fold way First by local contributions to the gaugecoupling constant via the one-loop threshold contributions ∆a These terms origi-nate from massive string states The second way to cancel the σ-model anomaliesis due to a field dependent (however gauge group independent) one-loop contribu-tion to the Kahler potential of the chiral moduli fields It implies that some of themoduli fields transform non-trivially under the Kahler transformations and alsounder reparametrisations in moduli space The universal one-loop modificationof the Kahler potential is nothing else than a generalised Green-Schwarz mecha-nism cancelling the σ-model anomalies This is analogous to the Green-Schwarzmechanism which cancels anomalies of physical U(1) gauge fields whereas theσ-model anomalies correspond to unphysical composite gauge connections Ef-fectively it means that the Green-Schwarz mechanism with respect to the σ-modelanomalies can be described by a non-holomorphic one-loop field redefinition ofthe associated tree-level moduli fields

As we will see in type IIA orientifold models these field redefinitions act onthe real parts of the dilaton field S as well as the complex structure moduli UJ

S rarr S + δGS(U T )

UJ rarr UJ + δGSJ (U T ) (55)

These redefined fields are those that determine the gauge coupling constants1g2astring at the string scale Recall that as the tree-level gauge couplings (24)are already gauge group dependent so are these one-loop corrections but theonly dependence arises due to the universal one-loop redefinition of the modulifields It is in this sense that we still call these one-loop corrections to the gaugecouplings universal Note that in heterotic string compactifications the σ-modelGreen-Schwarz mechanism only acts on the heterotic dilaton field

10

51 Holomorphic gauge couplings for toroidal models

In summary equation (51) is to be understood recursively which means that onecan insert the tree-level results into the last three terms of eq (51) In additionone also has to include the universal field redefinition eq (55) in 1g2astring inthe left hand side of eq (51) in order to get a complete matching of all termsin eq (51) as we will demonstrate for the aforementioned toroidal orbifold inthe following For N = 2 sectors the one-loop threshold corrections to the gaugecoupling constant indeed contain a holomorphic Wilsonian term f

(1)a whereas

for N = 1 sectors only the non-holomorphic piece ∆nha is present

Specifically the holomorphic gauge kinetic function can now be determinedby comparing the string theoretical formula (31) for the effective gauge couplingwith the field theoretical one (51) The first thing to notice are the differentcutoff scales appearing in the two formulas One needs to convert one into theother using

M2s

M2Pl

prop exp(2φ4) prop (S U1 U2 U3)minus 1

2 (56)

Here φ4 is the four dimensional dilaton and the complex structure moduli in thesupergravity basis can be expressed in terms of φ4 and the complex structuremoduli uI = RI2RI1 as

S =1

2πeminusφ4

1radicu1 u2 u3

UI =1

2πeminusφ4

radicuJ uKuI

with I 6= J 6= K 6= I (57)

These fields are the real parts of complex scalars of four dimensional chiral mul-tiplets Sc and U c

I As N = 4 super YangndashMills theory is finite one expects the sum of the terms

in (31) proportional to T (Ga) to cancel This is because the only chiral multipletstransforming in the adjoint representation of the gauge group are the open stringmoduli which (on the background considered) assemble themselves into threechiral multiplets thus forming an N = 4 sector together with the gauge fieldsTo show that this cancellation does happen one notices the following Firstlynadjoint = 3 as explained such that there is no term in ba proportional to T (Ga)Secondly

K = minus ln(Sc + Sc)minus

3sum

I=1

ln(U cI + U

c

I)minus3sum

I=1

ln(T cI + T

c

I) (58)

gminus2atree = S

3prod

I=1

nIa minus

3sum

I=1

UI nIam

Jam

Ka I 6= J 6= K 6= I (59)

where TI are the Kahler moduli of the torus and nIa m

Ia are the wrapping numbers

of the brane Finally one needs the matter metric for the open string moduli

11

which can be obtained from the T-dual expression in models with D9- and D5-branes [4] Performing the T-duality which essentially amounts to exchangingKahler and complex structure moduli and converting gauge flux into non-trivialintersection angles for the D6-branes one arrives at (I = 1 3)1

KIij =

δijTIUI

∣∣∣∣∣(nJ

a + iuJ mJa )(n

Ka + iuK m

Ka )

(nIa + iuI mI

a)

∣∣∣∣∣ I 6= J 6= K 6= I (510)

Let us now turn to the fields in the fundamental representation of the gaugegroup Ga in particular to the fields arising from the intersection with one otherstack of branes denoted by b For an N = 1 open string sector the metric forthese fields can be written as [34 4] (see also [35])

Kabij = δij S

minusα

3prod

I=1

Uminus(β+ξ θI

ab)

I Tminus(γ+ζ θI

ab)

I

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (511)

where α β γ ξ and ζ are undetermined constants As θ12ab gt 0 and θ3ab lt 0which is assumed in (511) the intersection number Iab is positive implying that

nf = IabNb (512)

Using Ta(f) =12and relations (52 56 58 511 512) one finds a contribution

to the right hand side of (51) proportional to

IabNb

2

(ln

(M2

s

micro2

)+ (2γ minus 1) ln(T1T2T3) + (2β minus 1

2) ln(U1U2U3) + (2αminus 1

2) lnS

+ζ

3sum

I=1

θIab lnTI + ξ

3sum

I=1

θIab lnUI minus ln

[Γ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab)

]) (513)

Using (36) one finds that the first and the last term exactly reproduce the contri-bution of the last two terms in (31) The terms proportional to ζ and ξ will laterbe shown to constitute the aforementioned universal gauge coupling correctionThe remaining three terms can neither be attributed to such a correction nor canthey be written as the real part of a holomorphic function Thus they cannot bethe one-loop correction to the gauge kinetic function and therefore must vanishThis fixes some of the coefficients in the ansatz (511)

α = β =1

4 γ =

1

2 (514)

1An overall factor involving the wrapping numbers was introduced in this expression in orderto achieve full cancellation This can be done as the expressions used are derived only up tooverall constants [34]

12

The same matching of terms appears between the Mobius diagram plus the annu-lus with boundaries on brane a and its orientifold image and the Kahler metricsfor fields in the symmetric and antisymmetric representation Here one has toreplace θIab and IabNb by θIaaprime = 2θIa and IaaprimeNa in (511) and (33) Apart fromthese replacements the Kahler metric for matter in these representations is alsogiven by (511) with the constants α β γ given in (514)

The corrections to the gauge couplings coming fromN = 2 open string sectorswere seen in the previous section to take on quite a different form They containa term

minus ln |η(i T cI )|4 = minus4Re [ln η(i T c

I )] (515)

which can be written as the real part of a holomorphic function This leads oneto conclude that the gauge kinetic function receives one-loop corrections fromthese sectors Inserting the correct prefactor which from the first term in (35)and the corresponding one in (51) can be seen to be proportional to the betafunction coefficient gives

f (1)a = minusNb |IJab IKab |

4π2ln η(i T c

I ) I 6= J 6= K 6= I (516)

where again I denotes the torus in which the branes lie on top of each other andIJKab are the intersection numbers on the other tori

The term minus ln(TI VaI ) in (35) is not the real part of a holomorphic function

Proceeding as before one finds that the Kahler metric for the hypermultiplet (ortwo chiral multiplets) living at an intersection of branes a and b preserving eightsupercharges must be

KIij =

|nIa + iuI m

Ia|

(UJ UK T J TK)12

I 6= J 6= K 6= I (517)

Apart from the factor in the numerator this is in agreement with the form foundby direct calculations [34 4] The appearance of the numerator is however plau-sible as it also appears in the open string moduli metric and the hypermultipletsunder discussion should feel the Irsquoth torus in the same way

52 Universal threshold corrections

In the following the aforementioned ldquouniversalrdquo gauge coupling corrections willbe discussed They also appear in the heterotic [13] and type I [36 37] string andare related to a redefinition of the dilaton at one-loop [9 8] This stems from thefact that the dilaton really lives in a linear multiplet rather than a chiral one

Our general ansatz for the Kahler metrics for the chiral matter in an N = 1sector contains a factor

3prod

J=1

Uminusξ θJ

ab

J Tminusζ θJ

ab

J (518)

13

which according to (51) appears in the one-loop correction to the gauge couplingconstant Neither is this term reproduced in the string one-loop calculation ofthe coupling nor can it be written as a correction to the holomorphic gaugekinetic function Therefore as is familiar from gauge threshold computationsthere remains the possibility that it can be absorbed into a one-loop correctionto the S and UI chiral superfields In the following we require that such a gaugegroup factor independent universal correction is possible and see how this fixesthe parameters in (518)

The first observation is that in order to get something gauge group indepen-dent the factor (518) actually must have the following form

3prod

J=1

Uminusξprime sign(Iab)θ

Jab

J Tminusζprime sign(Iab)θ

Jab

J (519)

with ξprime and ζ prime independent of the brane For the metrics of fields transformingin the symmetric or antisymmetric representation of the gauge group one hasto replace φab = φa minus φb by φaaprime = 2φa and sign(Iab) by sign(Iaaprime minus IaO6) orsign(Iaaprime + IaO6) respectively Then one computes (K prime denotes the factor (518519) appearing in the full Kahler metric K)

sum

r

Ta(r) ln detKprimer = |Iab|Nb

2ln

[3prod

J=1

Uminusξprime sign(Iab) θ

Jab

J Tminusζprime sign(Iab) θ

Jab

J

](520)

+|Iabprime |Nb

2ln

[3prod

J=1

Uminusξprime sign(Iabprime) θ

Jabprime

J Tminusζprime sign(Iabprime ) θ

Jabprime

J

]

+ Na+22

|IaaprimeminusIaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(IaaprimeminusIaO6) θ

Ja

J Tminus2ζprime sign(IaaprimeminusIaO6) θ

Ja

J

]

+ Naminus22

|Iaaprime+IaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(Iaaprime+IaO6) θ

Ja

J Tminus2ζprime sign(Iaaprime+IaO6) θ

Ja

J

]

After a few steps using |Iab| sign(Iab) = Iab and the tadpole cancellation condi-tion this can be brought to the simple form

sum

r

Ta(r) ln detKprimer = minusn1

an2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

minus3sum

J=1

nJam

Ka m

La

[sum

b

NbmJb n

Kb n

Lb

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

J 6= K 6= L 6= J (521)

Therefore these corrections have precisely the form required for them to be iden-tified with the one-loop correction between the linear superfields appearing in

14

string theory and the chiral superfields used in the supergravity description

SL = S minus 1

8π2

sum

b

Nbm1bm

2bm

3b

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

ULJ = UJ +

1

8π2

sum

b

NbmJb n

Kb n

Lb

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

J 6= K 6= L 6= J (522)

In contrast to eq (59) where the tree-level gauge couplings are determined theone-loop gauge couplings at the string scale have to include the redefined fieldsSL and UL

gminus2astring = SL

3prod

I=1

nIa minus

3sum

I=1

ULI n

Iam

Jam

Ka (523)

In contrast to all (to us) known cases studied in the literature for ξprime 6= 0 thefields which are corrected ie the moduli S and UI also appear in the one-loopredefinition Let us propose an argument why such corrections might be expectedto be absent Due to the anomalous U(1) gauge symmetries the chiral superfieldsS and UI participate in the Green-Schwarz mechanism and therefore transformnon-trivially under U(1) gauge transformations This implies that in order tobe gauge invariant the one-loop corrections in (522) proportional to lnUI mustbe extended in the usual way by UI rarr UI + δaGSVa Computing the resultingFI-terms via the supergravity formula ξa2g

2a = partKpartVa|Va=0 gives besides the

tree-level result depending on SL ULI a one-loop contribution proportional to

ξprimesum

b ξ(0)b φJ

b UJ This has an extra dependence on the complex structure moduliUI However for intersecting D6-branes we have seen that the Wilsonian (super-gravity) FI-Terms are proportional to the Wilsonian gauge threshold correctionswhich depend only on the Kahler moduli via instanton corrections (for N = 1sectors they are even vanishing in the setup at hand) This seems to suggest thatthere should better be no UminusθI

I dependence in the matter field Kahler metricsie ξprime = 0

Moreover in analogy to the heterotic string we expect that for the rangeminus1 le θJ le 1 the exponent of TJ runs over the range [minus1 0] This conditionwould fix ζ prime = plusmn12

Let us summarise the conclusions we have drawn from requiring holomorphyof the Wilsonian gauge kinetic function First we provided arguments that theKahler metric for N = 1 chiral matter fields in intersecting D6-brane models isof the following form

Kabij = δij S

minus 14

3prod

J=1

Uminus 1

4J T

minus( 12plusmn 1

2sign(Iab) θ

Jab)

J

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (524)

15

where supersymmetry of course requiressum3

I=1 θIab = 0 Second the holomorphic

gauge kinetic function (on the background considered) only receives correctionsfrom N = 2 open string sectors and the one-loop correction takes on the followingform

f (1)a = minus

sum

b

Nb |IJabIKab |4π2

ln η(i T cI ) I 6= J 6= K 6= I (525)

where the sum only runs over branes b which lie on top of brane a in exactly onetorus denoted by I Therefore the results for the gauge threshold corrections andthe matter field Kahler metrics are consistent both with the non-renormalisationtheorem from section 2 and the Kaplunovsky-Louis formula (51) Clearly itwould be interesting along the lines of [34] to carry out a string amplitude com-putation to fix the free coefficient in the ansatz (511) and see whether our indirectarguments are correct

6 Holomorphic E2-instanton amplitudes

Space-time instantons are given also by D-branes which in this case are EuclideanD2-branes (so-called E2-branes) wrapping three-cycles Ξ in the CalabindashYau sothat they are point-like in four-dimensional Minkowski space Such instantonscan contribute to the holomorphic superpotential and gauge kinetic functions onlyif they preserve half of the N = 1 supersymmetry This means that the instantonmeasure must contain a factor d4x d2θ Let us first clarify an important aspectof this half-BPS condition In the second part of this section we then revisitthe computation of contributions of such instantons to the superpotential andalso clarify some issues concerned with the appearing one-loop determinants Inthe third and fourth part we investigate under which conditions such stringinstantons can also contribute to the gauge kinetic functions and FI-terms

61 Half-BPS instantons

As has been explained in [24 18 20] just wrapping an E2-instanton around arigid sLag three-cycle in the Calabi-Yau gives four bosonic and four fermioniczero modes The vertex operators for the latter are

V(minus12)θ (z) = θα e

minusϕ(z)2 (z)Sα(z) Σh= 3

8q= 3

2(z) (61)

and

V(minus12)

θ(z) = θα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 3

2(z) (62)

Therefore if the instanton is not invariant under the orientifold projection onestill has four instead of the desired two fermionic zero modes Thus only by

16

placing the E2-brane in a position invariant under Ωσ does one have a chanceto get rid of the two additional zero modes θ For so called O(n) instantons onecan see that the zero modes xmicro θ are symmetrised and the mode θ gets anti-symmetrised For the opposite projection ie for USp(2n) instantons the zeromodes xmicro θ are anti-symmetrised and the mode θ gets symmetrised Thereforeone can only get the simple d4x d2θ instanton measure for a single O(1) instanton

62 Superpotential contributions

In order to contribute to the superpotential we also require that there do notarise any further zero modes from E2-E2 open strings so that the three-cycleΞ should be rigid ie b1(Ξ) = 0 Therefore considering an E2-instanton in anintersecting brane configuration additional zero modes can only arise from theintersection of the instanton Ξ with D6-branes Πa There are Na [ΞcapΠa]

+ chiralfermionic zero modes λaI and Na [Ξ cap Πa]

minus anti-chiral ones λaJ 2

For its presentation it is useful to introduce the short-hand notation

Φak bk [~xk] = Φak xk1middot Φxk1xk2

middot Φxk2xk3middot middot Φxknminus1xkn

middot Φxkn(k)bk (63)

for the chain-product of open string vertex operators Here we define Φak bk [~0] =Φak bk

To extract the superpotential one can probe it by evaluating an appropriatematter field correlator in the instanton background The CFT allows one tocompute it in physical normalisation which combines the superpotential part Ywith the matter field Kahler metrics like

〈Φa1b1 middot middot ΦaM bM 〉E2minusinst =e

K

2 YΦa1b1ΦaMbMradic

Ka1b1 middot middotKaM bM

(64)

In [14] a general expression for the single E2-instanton contribution to thecharged matter superpotential was proposed involving the evaluation of the fol-lowing zero mode integral over disc and one-loop open string CFT amplitudes

〈Φa1b1 middot middot ΦaM bM 〉E2 =V3gs

intd4x d2θ

sum

conf

proda

(prod[ΞcapΠa]+

i=1 dλia) (prod[ΞcapΠa]minus

i=1 dλi

a

)

exp(minusSE2) exp (Z prime0(E2)) 〈Φa1b1[~x1]〉λa1 λb1

middot middot 〈ΦaLbL[~xL]〉λaLλbL

(65)

For simplicity we do not consider the case that matter fields are also assignedto string loop diagrams The one-loop contributions are annulus diagrams for

2Here we introduced the physical intersection number between two branes Πa cap Πb whichis the sum of positive [Πa capΠb]

+ and negative [Πa cap Πb]minus intersections

17

open strings with one boundary on the E2-instanton and the other boundary onthe various D6-branes and Mobius diagrams with boundary on the E2-instanton

〈1〉1-loop = Z prime0(E2) =

sumb Z

primeA(E2aD6b) + Z primeM(E2aO6) (66)

Here Z prime means that we only sum over the massive open string states in theloop amplitude as the zero modes are taken care of explicitly It was shownthat these instantonic open string loop diagrams are identical to the one-loopthreshold corrections TA(D6aD6b) Diagrammatically we have the intriguingrelation shown in figure 1 and in figure 2 which holds for the even spin structures3

Fa

Fa x

x

D6aaE2 =D6b D6b

Figure 1 Relation between instantonic one-loop amplitudes and corresponding gaugethreshold corrections

Fa

Fa x

x

D6aaE2 =

O6O6

Figure 2 Relation between instantonic Mobius amplitude and corresponding gaugethreshold corrections

The annulus threshold corrections can be computed leading to

ZA(E2aD6b) =

int infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it)

η3(it)ACY

ab [αβ](it) (67)

and the Mobius strip amplitude for the instanton which as we explained mustbe invariant under the orientifold projection yields

ZM(E2aO6) = plusmnint infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it + 1

2

)

η3(it + 1

2

) ACYaa [α

β](it + 1

2

)(68)

3The contribution of the CP-odd Rminus sector is expected to yield corrections to the θ-angle

18

The overall plus sign is for O(1) instantons reflecting the fact that only for thesethe xmicro and θα zero modes survive the orientifold projection Note that up to theargument the Mobius thresholds are ZA(E2aD6a) Therefore for rigid branesthe massless sector reflects the number of four bosonic and two fermionic zeromodes In section 63 we will discuss the number of zero modes if b1(Ξ) gt 0 Allthese stringy threshold corrections are known to be non-holomorphic Thereforeit is not immediately obvious in which sense the expression (65) is meant andhow one can extract the holomorphic superpotential part Y from it

The CFT disc amplitudes in (65) are also not holomorphic but combinenon-holomorphic Kahler potential contributions and holomorphic superpotentialcontributions in the usual way [38 39 40 41]

〈Φab[~x]〉λaλb=

eK

2 YλaΦax1Φx1x2 ΦxN b λbradicKλaaKax1 KxnbKbλb

(69)

=e

K

2 YλabΦab[x]λbradic

Kλaa Kab[x]Kbλb

(610)

Due to the Kaplunovsky-Louis formula (51) the stringy one-loop amplitudes areknown to include the holomorphic Wilsonian part and contributions from wave-function normalisation Applied to the instanton one-loop amplitudes appearingin Z0(E2a) we write

Z0(E2a) = minus8π2 real(f (1)a )minus ba

2ln

(M2

p

micro2

)minus ca

2Ktree (611)

minus ln

(V3gs

)

tree

+sum

b

|IabNb|2

ln[detKab

]tree

where for the brane and instanton configuration in question the coefficients are

ba =sum

b

|IabNb|2

minus 3 ca =sum

b

|IabNb|2

minus 1 (612)

The constant contributions arise from the Mobius amplitude Inserting (69) and(611) in (65) one realises that the Kahler metrics involving an instanton zeromode and a matter field precisely cancel out so that only the matter metricssurvive as required by the general form (64) Moreover the term exp(K2)comes out just right due to the rule that each disc contains precisely two instantonzero modes The holomorphic piece in (64) can therefore be expressed entirelyin terms of other holomorphic quantities like holomorphic Yukawa couplingsthe holomorphic instanton action and the one-loop holomorphic Wilsonian gaugekinetic function on the E2-brane

YΦa1b1ΦaMbM

=sum

conf

signconf exp(minusSE2)tree exp(minusf (1)

a

)

Yλa1bΦa1b1

[~x1] λb1middot middot Yλa1

bΦaLbL[~xL]λbL

(613)

19

This explicitly shows that knowing the tree-level Kahler potentials computingthe matter field correlator in the instanton background up to one-loop level ings is sufficient to deduce the Wilsonian holomorphic instanton generated super-potential Higher order corrections in gs only come from loop corrections to theKahler potentials

63 Instanton corrections to the gauge kinetic functions

So far we have discussed space-time instanton corrections to the superpotentialThese involved one-loop determinants which are given by annulus vacuum di-agrams with at least one E2 boundary These are related to one-loop gaugethreshold corrections to the gauge theory on a D6-brane wrapping the same cycleas the E2 instanton

Now we can ask what other corrections these space-time instantons can induceBy applying S- and T-dualities to the story of world-sheet instanton correctionsin the heterotic string we expect that there can also be E2-instanton corrections

to the holomorphic gauge kinetic functions In the heterotic case similar to thetopological Type II string such corrections arise from string world-sheets of Eulercharacteristic zero ie here from world-sheets with two boundaries Thereforewe expect such corrections to appear for E2-instantons admitting one complexopen string modulus ie those wrapping a three-cycle with Betti number b1(Ξ) =1

Let us start by discussing the instanton zero mode structure for such a cycleFirst let us provide the form of the vertex operators The bosonic fields in the(minus1) ghost picture are

V (minus1)y (z) = y eminusϕ(z)Σh= 1

2q=plusmn1(z) (614)

which before the orientifold projection are accompanied by the two pairs offermionic zero modes

V (minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 1

2(z) (615)

and

V(minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=+ 1

2(z) (616)

Now one has to distinguish two cases depending on how the anti-holomorphicinvolution σ acts on the open string modulus Y

σ y rarr plusmny (617)

In the case that y is invariant under σ called first kind in the following theorientifold projection acts in the same way as for the 4D fields Xmicro ie thetwo bosonic zero modes y and the two fermionic zero modes micro survive In the

20

other case dubbed second kind the bosonic zero mode is projected out andonly the fermionic modulino zero mode micro survives4 Therefore in the absence ofany additional zero modes for instance from E2-D6 intersections the zero modemeasure in any instanton amplitude assumes the following form

intd4x d2θ d2y d2micro eminusSE2 for σ y rarr y (618)

andintd4x d2θ d2micro eminusSE2 for σ y rarr minusy (619)

As an example consider the set-up in figure 3 with σ yi rarr minusyi Here the

x x x2 31

1 2 3y y y

O6

E2

∆x1 ∆ y3

∆x2

Figure 3 Deformations of an instanton which is invariant under the orientifold pro-jection

deformations ∆x12 are of the first kind and ∆y3 is of the second kindNow it is clear that an instanton with precisely one set of fermionic zero

modes of the second kind and no additional zero modes can generate a correctionto the SU(Na) gauge kinetic function The instanton amplitude takes on thefollowing form

〈Fa(p1)Fa(p2)〉E2 =

intd4x d2θ d2micro exp(minusSE2) exp (Z prime

0(E2)) AF 2a(E2 D6a)

where AF 2a(E2 D6a) is the annulus diagram in figure 4 which absorbs all the

appearing fermionic zero modes and where the gauge boson vertex operators inthe (0)-ghost picture have the usual form

V(0)A (z) = ǫmicro (partmicroX(z) + i(p middot ψ)ψmicro(z)) e

ipmiddotX(z) (620)

4By duality this distinction is related to the two kinds of deformations of genus g curvesstudied in [42] The first kind are the curves moving in families ie transversal deformationof the curve The second kind is related to the deformations coming with the genus g of thecurve

21

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

persymmetric at tree-level in a globally consistent model no such correctionsare generated at one-loop We ask the question whether a small non-vanishingFI-term on a D6b-brane can induce an FI-term on a D6a-brane which is super-symmetric at tree-level We find the intriguing result that the one-loop inducedFI-term on brane D6a can be expressed by the gauge threshold corrections

Moving back to gauge couplings the extraction of the Wilsonian part involvesan interesting interplay between the non-holomorphic gauge couplings and theKahler potentials for all the matter fields involved providing strong constraintson the complete moduli dependence of the matter field Kahler potentials As wewill see in order to cancel all σ-model anomalies in the effective action a one-loop redefinition of the dilaton S-field as well as of the complex structure moduliU is needed The hereby induced corrections to the gauge coupling constants willbe referred to as ldquouniversalrdquo threshold corrections (in analogy with the heteroticstring) This is in contrast to perturbative heterotic string compactificationswhere only the dilaton acquires a universal one-loop field redefinition [8 9 1011 12 13]

Having revisited and discussed the perturbative one-loop corrections in thesecond part of the paper we undertake some first steps towards a better un-derstanding of possible D-instanton effects Such effects are very important foran understanding of the vacuum structure (these days called the landscape) ofstring compactifications and it has been pointed out recently that they can alsogenerate phenomenologically appealing terms like Majorana masses for neutrinos[14 15 16 17 18 19 20] Moreover they are also important for the string the-ory description of gauge instanton effects [21 22 23 24 25] (see [26] for a recentgeneral review on instantons)

String instantons are given by wrapped string world-sheets as well as by wrap-ped Euclidean D-branes and like in field theory their contributions to the space-time superpotential are quite restricted These contributions can be computed ina semi-classical approach ie one involving only the tree-level instanton actionand a one-loop determinant for the fluctuations around the instanton [27] Fortype IIA orientifold models on CalabindashYau spaces with intersecting D6-branes(and their T-dual cousins) the contribution of wrapped Euclidean D2-braneshereafter called E2-branes to the superpotential has been determined in [14] (seealso [15]) Since both the D6-branes and the E2-instantons are described by openstring theories it was shown that (in the spirit of the D(minus1) instantons treatedeg in [28 29 21]) the entire instanton computation boils down to the evaluationof disc and one-loop string diagrams with boundary (changing) operators insertedHere both the D6-branes and the E2-instantons wrap compact three-cycles of theCalabindashYau manifold

Intriguingly the one-loop contributions in the instanton amplitude [14] havebeen shown to be identical to string threshold corrections for the gauge couplingsof the corresponding D6-branes [17 23] This relates the computation of suchinstanton amplitudes to the discussion in the first half of this paper So far it has

3

not been explained explicitly in which sense the computed instantonic correlationfunctions are meant to be holomorphic With the results from the first part ofthis paper we clarify this point

Finally we show that E2-instantons not only contribute to the superpotentialbut from the zero mode counting can also contribute to the holomorphic gaugekinetic functions for the SU(N) gauge groups localised on the D6-branes Inorder for such corrections to arise the E2-instanton must not be rigid but mustadmit one extra pair of fermionic zero modes arising from a deformation of theinstanton This is the space-time instanton generalisation of a fact known fromtopological string theory namely that world-sheet instantons induce tr(W 2)hminus1

couplings if they have h boundaries We will see that such couplings can also arisefrom space-time E2-instantons Finally we find that the zero mode counting alsoallows E2-instanton corrections to the FI-terms on the D6-branes Similar to theone-loop corrections these can arise once the supersymmetry on the E2-braneis softly broken by for instance turning on the C3-form modulus through theworld-volume of E2

2 A non-renormalisation theorem

Let us investigate the structure of perturbative and non-perturbative correctionsto the holomorphic gauge kinetic functions for Type II orientifolds We discussthis for Type IIA orientifolds but this is of course related via mirror symmetryto the corresponding Type IIB orientifolds

Consider a Type IIA orientifold with O6-planes and intersecting D6-branespreserving N = 1 supersymmetry in four dimensions ie the D6-branes wrapspecial Lagrangian (sLag) three-cycles Πa of the underlying CalabindashYau manifoldX all preserving the same supersymmetry On the threefold we introduce in theusual way a symplectic basis (AI B

I) I = 0 1 h21 of homological three-cycles with the topological intersection numbers

AI BJ = δJI (21)

Moreover we assume that the AI cycles are invariant under the orientifold pro-jection and that the BJ cycles are projected out The complexified complexstructure moduli on such an orientifold are defined as

U cI =

1

(2π) ℓ3s

[eminusφ4

int

AI

real(Ω3) minus i

int

AI

C3

] (22)

where Ω3 denotes the normalised holomorphic three-form on X and the four-dimensional dilaton is defined by φ4 = φ10 minus 1

2ln(VXℓ

6s) Expanding a three

cycle Πa into the symplectic basis

Πa =M Ia AI +NaI B

I (23)

4

with M I NI isin Z from dimensional reduction of the Dirac-Born-Infeld (DBI)action one can deduce the SU(Na) gauge kinetic functions at string tree-level

fa =

h21sum

I=0

M Ia U

cI (24)

Since the imaginary parts of the U cI are axionic fields they enjoy a Peccei-Quinn

shift symmetry U cI rarr U c

I + cI which is preserved perturbatively and only brokenby E2-brane instantons

Let Ci denote a basis of anti-invariant 2-cycles ie Ci isin H11minus The complex-

ified Kahler moduli are then defined as

T ci =

1

ℓ2s

(int

Ci

J2 minus i

int

Ci

B2

) (25)

where B2 denotes the NS-NS two-form of the Type IIA superstring Thereforealso the complexified Kahler moduli enjoy a Peccei-Quinn shift symmetry brokenby world-sheet instantons Note that the chiral fields T c

i organise the σ-modelperturbation theory and do not contain the dilaton so that the string pertur-bative theory is entirely defined by powers of the U c

I Moreover to shorten thenotation we denote by U c

I and T ci the complexified moduli and by UI and Ti only

the real partsThe superpotentialW and the gauge kinetic function f in the four-dimensional

effective supergravity action are holomorphic quantities In the usual way em-ploying holomorphy and the Peccei-Quinn symmetries above one arrives at thefollowing two non-renormalisation theorems

The superpotential can only have the following dependence on U cI and T c

i

W = Wtree +W np(eminusUc

I eminusT ci

) (26)

ie beyond tree-level there can only be non-perturbative contributions fromworld-sheet and E2-brane instantons Similarly the holomorphic gauge kineticfunction must look like

fa =sum

I

M IaU

cI + f 1-loop

a

(eminusT c

i

)+ fnp

a

(eminusUc

I eminusT ci

) (27)

ie in particular its one-loop correction must not depend on the complex structuremoduli Finally we consider the Fayet-Iliopoulos terms for the U(1)a gaugefields on the D6-branes At string tree-level and for small deviations from thesupersymmetry locus these are given by

ξa = eminusφ4

int

Πa

image(Ω3) = eminusφ4 N Ia

int

BI

image(Ω3) (28)

5

and therefore only depend on the complex structure moduli At this classicallevel there are no αprime corrections It is an important question about brane stabilitywhether these FI-terms receive perturbative or non-perturbative corrections in gsAgain non-renormalisation theorems say that in the Wilsonian sense one expectsperturbative corrections at most at one-loop

In the following we will be concerned with the terms beyond tree-level appear-ing in (26) (27) and for the FI-terms First we discuss the one-loop thresholdcorrections f 1minusloop

(eminusT c

i

) which also make their appearance in the space-time

instanton generated superpotential W np(eminusUc

I eminusT ci

) Second we will revisit the

computation of stringy one-loop corrections to the FI-terms Finally we willdiscuss fnp

(eminusUc

I eminusT ci

)as well as instanton corrections to the FI-terms

3 One-loop thresholds for intersecting D6-branes

on T6

The purpose of this section is to recall the one-loop results for the gauge thresh-old corrections in intersecting D6-brane models [5 30 6] The gauge couplingconstants of the various gauge group factors Ga in such a model up to one loophave the form

8π2

g2a(micro)=

8π2

g2astring+ba2

ln

(M2

s

micro2

)+

∆a

2 (31)

where ba is the beta function coefficient The first term corresponds to the gaugecoupling constant at the string scale which contains the tree-level gauge couplingas well as the ldquouniversalrdquo contributions at one-loop (see section 52) Thesecontributions are universal in the sense that they originate from a redefinition ofthe dilaton and complex structure moduli at one-loop The redefinition is branestack and therefore gauge group independent However as the gauge couplingsdiffer for the various gauge groups already at tree level this correction effectivelyis gauge group dependent The second term gives the usual one-loop running ofthe coupling constants and the third term denotes the one-loop string thresholdcorrections originating from integrating out massive string excitations The lasttwo terms can be computed as a sum of all annulus and Mobius diagrams withone boundary on brane a in the presence of a background magnetic field in thefour-dimensional space-time

ba ln

(M2

s

micro2

)+∆a =

sum

b

TA(D6aD6b) +sum

bprime

TA(D6aD6bprime) (32)

+ TA(D6aD6aprime) + TM(D6aO6)

Here D6cprime denotes the orientifold image of brane c In an orbifold one also hasto take into account the orbifold images of the branes and orientifold planes

6

The relevant amplitudes for the Z2timesZ2 orbifold have been computed [5 6] Ina sector preserving N = 1 supersymmetry (this means in particular

sumI θ

Iab = 0)

the annulus and Mobius amplitudes are (after subtracting terms which uponsumming over all diagrams vanish due to the tadpole cancellation condition) [6]

TA(D6aD6b) =IabNb

2

[ln

(M2

s

micro2

) 3sum

I=1

sign(θIab)minus

minus ln3prod

I=1

(Γ(|θIab|)

Γ(1minus |θIab|)

)sign(θIab)

minus3sum

I=1

sign(θIab) (ln 2minus γ)

] (33)

TM(D6aO6k) = plusmnIaO6k

[ln

(M2

s

micro2

) 3sum

I=1

sign(θIaO6k)minus

minus ln3prod

I=1

(Γ(2|θIaO6k

|)Γ(1minus 2|θIaO6k

|)

)sign(θIaO6k

)

+3sum

I=1

sign(θIaO6k)(γ minus 3 ln 2)

] (34)

where Iab is the intersection number of branes a and b Nb is the number ofbranes on stack b and πθIab is the intersection angle of branes a and b on the Irsquothtorus Similarly IaO6k denotes the intersection number of brane a and orientifoldplane k and πθIaO6k

their intersection angle The formula for TM is only valid for|θIaO6k

| lt 12 the formulas for other cases look similar [6]In a sector preserving N = 2 supersymmetry one finds [5]

TA(D6aD6b) = Nb|IJab IKab |[ln

(M2

s

micro2

)minus ln |η(i T c

I )|4 minus ln(TI VaI ) + γE minus ln(4π)

]

(35)where I denotes the torus on which the branes lie on top of each other TI itsKahler modulus and T c

I its complexification with TI = real(T cI ) Furthermore

V aI = |nI

a + iuImIa|2uI with uI the complex structure modulus of the torus and

nIam

Ia the wrapping numbers on the Irsquoth torus Note that the moduli dependence

of the one-loop threshold function in the N = 2 sectors is in complete agreementwith the non-renormalisation theorems of section two (see eq (27)) since theholomorphic part of ∆N=2

a is proportional only to ln η(i T cI )

For theN = 1 sectors the one-loop thresholds in a given open string D6-branesector have the following form (specialising to the case θ12ab gt 0 θ3ab lt 0)

∆a = minus ba16π2

ln

[Γ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab)

] (36)

This expression is a non-holomorphic function of the complex structure moduliU cI Hence for the N = 1 sectors the holomorphic one-loop gauge kinetic func-

tion f 1minusloopa

(eminusT c

i

)vanishes The emergence of the non-holomorphic terms in the

one-loop threshold corrections will be further discussed in section 5

7

4 Fayet-Iliopoulos terms

In this section we investigate one-loop corrections to the FI-terms for a U(1)agauge field on the D6a-brane induced by the presence of other branes D6b Suchcorrections for Type I string vacua have already been studied in [31 32] and thecase of intersecting D6-branes has been discussed in [7] Here we are followingessentially the computational technique of [7] The crucial observation is thatthe vertex operator for the auxiliary D-field in the (0)-ghost picture is simply

given by the internal world-sheet U(1) current ie V(0)Da

= JU(1) Therefore the

one-point function of V(0)Da

on the annulus with boundaries a and b can be writtenas

〈VDa〉 = minus i

2πpartν

int infin

0

dt Zab(ν it)|ν=0 (41)

where Zab(ν it) denotes the annulus partition function with insertion of exp(2πiJ0)in the open string sector (ab) where J0 is the zero mode of the U(1) current Inthe case of intersecting D6-branes on a torus preserving N = 1 supersymmetryand after application of the Riemann theta-identities this partition function isgiven by

Zab(ν it) = IabNb(minusi)3π4t2

ϑ1(3ν2 it)

prodI ϑ1(minusν

2+ iθI

2t it)

η3(it)prod

I ϑ1(iθI2t it)

(42)

Using that ϑ1(0 it) = 0 and ϑprime1(ν it)|ν=0 = minus2πη3 one obtains the divergence

〈VDa〉 ≃ IabNb

int infin

0

dt

t2 (43)

which is cancelled by tadpole cancellation in a global model Therefore oncethe D6-branes are supersymmetric at tree-level no FI-term is generated at one-loop level and the system is not destabilised This result is consistent with thecomputations in [31 32]

However this is not the end of the story of the one-loop corrections to FI-terms in intersecting D6-brane models One can also envision that a tree-levelFI-term on a brane D6b induces via a one-loop diagram an FI-term on a braneD6a To proceed we assume that θIb rarr θIb + 2ǫI with

sumǫI = ǫ and compute

〈VDa〉ǫI = minus i

2πpartνpartǫI

int infin

0

dt Zab(ǫI ν it)|ν=0ǫI=0 (44)

with

Zab(ǫI ν it) = IabNb

(minusi)3π4t2

ϑ1(3ν2+ i ǫ

2t it)

prodI ϑ1(minusν

2+ i (ǫ

IminusǫJminusǫK)2

t+ iθI2t it)

η3(it)prod

I ϑ1(iǫIt+ iθI

2t it)

8

The derivative with respect to the supersymmetry breaking parameters ǫI bringsdown one factor of t and it turns out that the result is the same for all partǫI

〈VDa〉ǫ ≃ i IabNb

int infin

0

dt

t

3sum

I=1

ϑprime1ϑ1

(iθI

abt

2 it2

) (45)

This is the same expression as the one-loop threshold corrections TA(D6a D6b)so that at linear order in ǫ we obtain for the FI-terms

(ξa)ǫ = (ǫb minus ǫa) TA(D6a D6b) (46)

Completely analogously one can show that this formula remains true also forN = 2 open string sectors Therefore we would like to propose that such arelation between gauge threshold and one-loop corrections to FI-terms is validfor general intersecting D6-brane models Moreover the Wilsonian part of thethresholds TA(D6a D6b) which we compute in the next section should also bethe Wilsonian part of the correction to the FI-terms

5 Wilsonian gauge kinetic function and σ-model

anomalies

In a supersymmetric gauge theory one can compute the running gauge couplingsga(micro

2) in terms of the gauge kinetic functions fa the Kahler potential K and theKahler metrics of the charged matter fields Kab(micro2) [33 8 9 10 11 12 13]

8π2

g2a(micro2)

= 8π2real(fa) +ba2

ln

(Λ2

micro2

)+ca2K

+T (Ga) ln gminus2a (micro2)minus

sum

r

Ta(r) ln detKr(micro2) (51)

with

ba =sum

r

nrTa(r)minus 3 T (Ga) ca =sum

r

nrTa(r)minus T (Ga) (52)

and Ta(r) = Tr(T 2(a)) (T(a) being the generators of the gauge group Ga) In addi-

tion T (Ga) = Ta(adjGa) and nr is the number of multiplets in the representationr of the gauge group and the sums run over these representations In this contextthe natural cutoff scale for a field theory is the Planck scale ie Λ2 =M2

PlThe left hand side of eq (51) is given by eq (31) which contains the gauge

coupling at the string scale 1g2astring as well as the one-loop string thresholdcorrections ∆a In general ∆a is the sum of a non-holomorphic term plus thereal-part of a holomorphic threshold correction

∆a = ∆nha + real(∆hol

a ) (53)

9

On the right hand side of eq (51) fa denotes the Wilsonian ie holomorphicgauge kinetic function which is given in terms of a holomorphic tree-level functionplus the holomorphic part of the one-loop threshold corrections (cf eq (27))

fa = f treea + f 1minusloop

a

(eminusT c

i

)=sum

I

M IaU

cI +∆hol

a (54)

In addition on the right hand side of eq (51) the non-holomorphic terms propor-tional to the Kahler metric of the moduli K and the matter field Kahler metricsKr are due to the one-loop contributions of massless fields These fields generatenon-local terms in the one-loop effective action which correspond to one-loopnon-invariances under σ-model transformations the so-called σ-model anomalies(Kahler and reparametrisation anomalies)

Matching up all terms in eq (51) essentially means that the σ-model anoma-lies can be cancelled in a two-fold way First by local contributions to the gaugecoupling constant via the one-loop threshold contributions ∆a These terms origi-nate from massive string states The second way to cancel the σ-model anomaliesis due to a field dependent (however gauge group independent) one-loop contribu-tion to the Kahler potential of the chiral moduli fields It implies that some of themoduli fields transform non-trivially under the Kahler transformations and alsounder reparametrisations in moduli space The universal one-loop modificationof the Kahler potential is nothing else than a generalised Green-Schwarz mecha-nism cancelling the σ-model anomalies This is analogous to the Green-Schwarzmechanism which cancels anomalies of physical U(1) gauge fields whereas theσ-model anomalies correspond to unphysical composite gauge connections Ef-fectively it means that the Green-Schwarz mechanism with respect to the σ-modelanomalies can be described by a non-holomorphic one-loop field redefinition ofthe associated tree-level moduli fields

As we will see in type IIA orientifold models these field redefinitions act onthe real parts of the dilaton field S as well as the complex structure moduli UJ

S rarr S + δGS(U T )

UJ rarr UJ + δGSJ (U T ) (55)

These redefined fields are those that determine the gauge coupling constants1g2astring at the string scale Recall that as the tree-level gauge couplings (24)are already gauge group dependent so are these one-loop corrections but theonly dependence arises due to the universal one-loop redefinition of the modulifields It is in this sense that we still call these one-loop corrections to the gaugecouplings universal Note that in heterotic string compactifications the σ-modelGreen-Schwarz mechanism only acts on the heterotic dilaton field

10

51 Holomorphic gauge couplings for toroidal models

In summary equation (51) is to be understood recursively which means that onecan insert the tree-level results into the last three terms of eq (51) In additionone also has to include the universal field redefinition eq (55) in 1g2astring inthe left hand side of eq (51) in order to get a complete matching of all termsin eq (51) as we will demonstrate for the aforementioned toroidal orbifold inthe following For N = 2 sectors the one-loop threshold corrections to the gaugecoupling constant indeed contain a holomorphic Wilsonian term f

(1)a whereas

for N = 1 sectors only the non-holomorphic piece ∆nha is present

Specifically the holomorphic gauge kinetic function can now be determinedby comparing the string theoretical formula (31) for the effective gauge couplingwith the field theoretical one (51) The first thing to notice are the differentcutoff scales appearing in the two formulas One needs to convert one into theother using

M2s

M2Pl

prop exp(2φ4) prop (S U1 U2 U3)minus 1

2 (56)

Here φ4 is the four dimensional dilaton and the complex structure moduli in thesupergravity basis can be expressed in terms of φ4 and the complex structuremoduli uI = RI2RI1 as

S =1

2πeminusφ4

1radicu1 u2 u3

UI =1

2πeminusφ4

radicuJ uKuI

with I 6= J 6= K 6= I (57)

These fields are the real parts of complex scalars of four dimensional chiral mul-tiplets Sc and U c

I As N = 4 super YangndashMills theory is finite one expects the sum of the terms

in (31) proportional to T (Ga) to cancel This is because the only chiral multipletstransforming in the adjoint representation of the gauge group are the open stringmoduli which (on the background considered) assemble themselves into threechiral multiplets thus forming an N = 4 sector together with the gauge fieldsTo show that this cancellation does happen one notices the following Firstlynadjoint = 3 as explained such that there is no term in ba proportional to T (Ga)Secondly

K = minus ln(Sc + Sc)minus

3sum

I=1

ln(U cI + U

c

I)minus3sum

I=1

ln(T cI + T

c

I) (58)

gminus2atree = S

3prod

I=1

nIa minus

3sum

I=1

UI nIam

Jam

Ka I 6= J 6= K 6= I (59)

where TI are the Kahler moduli of the torus and nIa m

Ia are the wrapping numbers

of the brane Finally one needs the matter metric for the open string moduli

11

which can be obtained from the T-dual expression in models with D9- and D5-branes [4] Performing the T-duality which essentially amounts to exchangingKahler and complex structure moduli and converting gauge flux into non-trivialintersection angles for the D6-branes one arrives at (I = 1 3)1

KIij =

δijTIUI

∣∣∣∣∣(nJ

a + iuJ mJa )(n

Ka + iuK m

Ka )

(nIa + iuI mI

a)

∣∣∣∣∣ I 6= J 6= K 6= I (510)

Let us now turn to the fields in the fundamental representation of the gaugegroup Ga in particular to the fields arising from the intersection with one otherstack of branes denoted by b For an N = 1 open string sector the metric forthese fields can be written as [34 4] (see also [35])

Kabij = δij S

minusα

3prod

I=1

Uminus(β+ξ θI

ab)

I Tminus(γ+ζ θI

ab)

I

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (511)

where α β γ ξ and ζ are undetermined constants As θ12ab gt 0 and θ3ab lt 0which is assumed in (511) the intersection number Iab is positive implying that

nf = IabNb (512)

Using Ta(f) =12and relations (52 56 58 511 512) one finds a contribution

to the right hand side of (51) proportional to

IabNb

2

(ln

(M2

s

micro2

)+ (2γ minus 1) ln(T1T2T3) + (2β minus 1

2) ln(U1U2U3) + (2αminus 1

2) lnS

+ζ

3sum

I=1

θIab lnTI + ξ

3sum

I=1

θIab lnUI minus ln

[Γ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab)

]) (513)

Using (36) one finds that the first and the last term exactly reproduce the contri-bution of the last two terms in (31) The terms proportional to ζ and ξ will laterbe shown to constitute the aforementioned universal gauge coupling correctionThe remaining three terms can neither be attributed to such a correction nor canthey be written as the real part of a holomorphic function Thus they cannot bethe one-loop correction to the gauge kinetic function and therefore must vanishThis fixes some of the coefficients in the ansatz (511)

α = β =1

4 γ =

1

2 (514)

1An overall factor involving the wrapping numbers was introduced in this expression in orderto achieve full cancellation This can be done as the expressions used are derived only up tooverall constants [34]

12

The same matching of terms appears between the Mobius diagram plus the annu-lus with boundaries on brane a and its orientifold image and the Kahler metricsfor fields in the symmetric and antisymmetric representation Here one has toreplace θIab and IabNb by θIaaprime = 2θIa and IaaprimeNa in (511) and (33) Apart fromthese replacements the Kahler metric for matter in these representations is alsogiven by (511) with the constants α β γ given in (514)

The corrections to the gauge couplings coming fromN = 2 open string sectorswere seen in the previous section to take on quite a different form They containa term

minus ln |η(i T cI )|4 = minus4Re [ln η(i T c

I )] (515)

which can be written as the real part of a holomorphic function This leads oneto conclude that the gauge kinetic function receives one-loop corrections fromthese sectors Inserting the correct prefactor which from the first term in (35)and the corresponding one in (51) can be seen to be proportional to the betafunction coefficient gives

f (1)a = minusNb |IJab IKab |

4π2ln η(i T c

I ) I 6= J 6= K 6= I (516)

where again I denotes the torus in which the branes lie on top of each other andIJKab are the intersection numbers on the other tori

The term minus ln(TI VaI ) in (35) is not the real part of a holomorphic function

Proceeding as before one finds that the Kahler metric for the hypermultiplet (ortwo chiral multiplets) living at an intersection of branes a and b preserving eightsupercharges must be

KIij =

|nIa + iuI m

Ia|

(UJ UK T J TK)12

I 6= J 6= K 6= I (517)

Apart from the factor in the numerator this is in agreement with the form foundby direct calculations [34 4] The appearance of the numerator is however plau-sible as it also appears in the open string moduli metric and the hypermultipletsunder discussion should feel the Irsquoth torus in the same way

52 Universal threshold corrections

In the following the aforementioned ldquouniversalrdquo gauge coupling corrections willbe discussed They also appear in the heterotic [13] and type I [36 37] string andare related to a redefinition of the dilaton at one-loop [9 8] This stems from thefact that the dilaton really lives in a linear multiplet rather than a chiral one

Our general ansatz for the Kahler metrics for the chiral matter in an N = 1sector contains a factor

3prod

J=1

Uminusξ θJ

ab

J Tminusζ θJ

ab

J (518)

13

which according to (51) appears in the one-loop correction to the gauge couplingconstant Neither is this term reproduced in the string one-loop calculation ofthe coupling nor can it be written as a correction to the holomorphic gaugekinetic function Therefore as is familiar from gauge threshold computationsthere remains the possibility that it can be absorbed into a one-loop correctionto the S and UI chiral superfields In the following we require that such a gaugegroup factor independent universal correction is possible and see how this fixesthe parameters in (518)

The first observation is that in order to get something gauge group indepen-dent the factor (518) actually must have the following form

3prod

J=1

Uminusξprime sign(Iab)θ

Jab

J Tminusζprime sign(Iab)θ

Jab

J (519)

with ξprime and ζ prime independent of the brane For the metrics of fields transformingin the symmetric or antisymmetric representation of the gauge group one hasto replace φab = φa minus φb by φaaprime = 2φa and sign(Iab) by sign(Iaaprime minus IaO6) orsign(Iaaprime + IaO6) respectively Then one computes (K prime denotes the factor (518519) appearing in the full Kahler metric K)

sum

r

Ta(r) ln detKprimer = |Iab|Nb

2ln

[3prod

J=1

Uminusξprime sign(Iab) θ

Jab

J Tminusζprime sign(Iab) θ

Jab

J

](520)

+|Iabprime |Nb

2ln

[3prod

J=1

Uminusξprime sign(Iabprime) θ

Jabprime

J Tminusζprime sign(Iabprime ) θ

Jabprime

J

]

+ Na+22

|IaaprimeminusIaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(IaaprimeminusIaO6) θ

Ja

J Tminus2ζprime sign(IaaprimeminusIaO6) θ

Ja

J

]

+ Naminus22

|Iaaprime+IaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(Iaaprime+IaO6) θ

Ja

J Tminus2ζprime sign(Iaaprime+IaO6) θ

Ja

J

]

After a few steps using |Iab| sign(Iab) = Iab and the tadpole cancellation condi-tion this can be brought to the simple form

sum

r

Ta(r) ln detKprimer = minusn1

an2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

minus3sum

J=1

nJam

Ka m

La

[sum

b

NbmJb n

Kb n

Lb

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

J 6= K 6= L 6= J (521)

Therefore these corrections have precisely the form required for them to be iden-tified with the one-loop correction between the linear superfields appearing in

14

string theory and the chiral superfields used in the supergravity description

SL = S minus 1

8π2

sum

b

Nbm1bm

2bm

3b

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

ULJ = UJ +

1

8π2

sum

b

NbmJb n

Kb n

Lb

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

J 6= K 6= L 6= J (522)

In contrast to eq (59) where the tree-level gauge couplings are determined theone-loop gauge couplings at the string scale have to include the redefined fieldsSL and UL

gminus2astring = SL

3prod

I=1

nIa minus

3sum

I=1

ULI n

Iam

Jam

Ka (523)

In contrast to all (to us) known cases studied in the literature for ξprime 6= 0 thefields which are corrected ie the moduli S and UI also appear in the one-loopredefinition Let us propose an argument why such corrections might be expectedto be absent Due to the anomalous U(1) gauge symmetries the chiral superfieldsS and UI participate in the Green-Schwarz mechanism and therefore transformnon-trivially under U(1) gauge transformations This implies that in order tobe gauge invariant the one-loop corrections in (522) proportional to lnUI mustbe extended in the usual way by UI rarr UI + δaGSVa Computing the resultingFI-terms via the supergravity formula ξa2g

2a = partKpartVa|Va=0 gives besides the

tree-level result depending on SL ULI a one-loop contribution proportional to

ξprimesum

b ξ(0)b φJ

b UJ This has an extra dependence on the complex structure moduliUI However for intersecting D6-branes we have seen that the Wilsonian (super-gravity) FI-Terms are proportional to the Wilsonian gauge threshold correctionswhich depend only on the Kahler moduli via instanton corrections (for N = 1sectors they are even vanishing in the setup at hand) This seems to suggest thatthere should better be no UminusθI

I dependence in the matter field Kahler metricsie ξprime = 0

Moreover in analogy to the heterotic string we expect that for the rangeminus1 le θJ le 1 the exponent of TJ runs over the range [minus1 0] This conditionwould fix ζ prime = plusmn12

Let us summarise the conclusions we have drawn from requiring holomorphyof the Wilsonian gauge kinetic function First we provided arguments that theKahler metric for N = 1 chiral matter fields in intersecting D6-brane models isof the following form

Kabij = δij S

minus 14

3prod

J=1

Uminus 1

4J T

minus( 12plusmn 1

2sign(Iab) θ

Jab)

J

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (524)

15

where supersymmetry of course requiressum3

I=1 θIab = 0 Second the holomorphic

gauge kinetic function (on the background considered) only receives correctionsfrom N = 2 open string sectors and the one-loop correction takes on the followingform

f (1)a = minus

sum

b

Nb |IJabIKab |4π2

ln η(i T cI ) I 6= J 6= K 6= I (525)

where the sum only runs over branes b which lie on top of brane a in exactly onetorus denoted by I Therefore the results for the gauge threshold corrections andthe matter field Kahler metrics are consistent both with the non-renormalisationtheorem from section 2 and the Kaplunovsky-Louis formula (51) Clearly itwould be interesting along the lines of [34] to carry out a string amplitude com-putation to fix the free coefficient in the ansatz (511) and see whether our indirectarguments are correct

6 Holomorphic E2-instanton amplitudes

Space-time instantons are given also by D-branes which in this case are EuclideanD2-branes (so-called E2-branes) wrapping three-cycles Ξ in the CalabindashYau sothat they are point-like in four-dimensional Minkowski space Such instantonscan contribute to the holomorphic superpotential and gauge kinetic functions onlyif they preserve half of the N = 1 supersymmetry This means that the instantonmeasure must contain a factor d4x d2θ Let us first clarify an important aspectof this half-BPS condition In the second part of this section we then revisitthe computation of contributions of such instantons to the superpotential andalso clarify some issues concerned with the appearing one-loop determinants Inthe third and fourth part we investigate under which conditions such stringinstantons can also contribute to the gauge kinetic functions and FI-terms

61 Half-BPS instantons

As has been explained in [24 18 20] just wrapping an E2-instanton around arigid sLag three-cycle in the Calabi-Yau gives four bosonic and four fermioniczero modes The vertex operators for the latter are

V(minus12)θ (z) = θα e

minusϕ(z)2 (z)Sα(z) Σh= 3

8q= 3

2(z) (61)

and

V(minus12)

θ(z) = θα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 3

2(z) (62)

Therefore if the instanton is not invariant under the orientifold projection onestill has four instead of the desired two fermionic zero modes Thus only by

16

placing the E2-brane in a position invariant under Ωσ does one have a chanceto get rid of the two additional zero modes θ For so called O(n) instantons onecan see that the zero modes xmicro θ are symmetrised and the mode θ gets anti-symmetrised For the opposite projection ie for USp(2n) instantons the zeromodes xmicro θ are anti-symmetrised and the mode θ gets symmetrised Thereforeone can only get the simple d4x d2θ instanton measure for a single O(1) instanton

62 Superpotential contributions

In order to contribute to the superpotential we also require that there do notarise any further zero modes from E2-E2 open strings so that the three-cycleΞ should be rigid ie b1(Ξ) = 0 Therefore considering an E2-instanton in anintersecting brane configuration additional zero modes can only arise from theintersection of the instanton Ξ with D6-branes Πa There are Na [ΞcapΠa]

+ chiralfermionic zero modes λaI and Na [Ξ cap Πa]

minus anti-chiral ones λaJ 2

For its presentation it is useful to introduce the short-hand notation

Φak bk [~xk] = Φak xk1middot Φxk1xk2

middot Φxk2xk3middot middot Φxknminus1xkn

middot Φxkn(k)bk (63)

for the chain-product of open string vertex operators Here we define Φak bk [~0] =Φak bk

To extract the superpotential one can probe it by evaluating an appropriatematter field correlator in the instanton background The CFT allows one tocompute it in physical normalisation which combines the superpotential part Ywith the matter field Kahler metrics like

〈Φa1b1 middot middot ΦaM bM 〉E2minusinst =e

K

2 YΦa1b1ΦaMbMradic

Ka1b1 middot middotKaM bM

(64)

In [14] a general expression for the single E2-instanton contribution to thecharged matter superpotential was proposed involving the evaluation of the fol-lowing zero mode integral over disc and one-loop open string CFT amplitudes

〈Φa1b1 middot middot ΦaM bM 〉E2 =V3gs

intd4x d2θ

sum

conf

proda

(prod[ΞcapΠa]+

i=1 dλia) (prod[ΞcapΠa]minus

i=1 dλi

a

)

exp(minusSE2) exp (Z prime0(E2)) 〈Φa1b1[~x1]〉λa1 λb1

middot middot 〈ΦaLbL[~xL]〉λaLλbL

(65)

For simplicity we do not consider the case that matter fields are also assignedto string loop diagrams The one-loop contributions are annulus diagrams for

2Here we introduced the physical intersection number between two branes Πa cap Πb whichis the sum of positive [Πa capΠb]

+ and negative [Πa cap Πb]minus intersections

17

open strings with one boundary on the E2-instanton and the other boundary onthe various D6-branes and Mobius diagrams with boundary on the E2-instanton

〈1〉1-loop = Z prime0(E2) =

sumb Z

primeA(E2aD6b) + Z primeM(E2aO6) (66)

Here Z prime means that we only sum over the massive open string states in theloop amplitude as the zero modes are taken care of explicitly It was shownthat these instantonic open string loop diagrams are identical to the one-loopthreshold corrections TA(D6aD6b) Diagrammatically we have the intriguingrelation shown in figure 1 and in figure 2 which holds for the even spin structures3

Fa

Fa x

x

D6aaE2 =D6b D6b

Figure 1 Relation between instantonic one-loop amplitudes and corresponding gaugethreshold corrections

Fa

Fa x

x

D6aaE2 =

O6O6

Figure 2 Relation between instantonic Mobius amplitude and corresponding gaugethreshold corrections

The annulus threshold corrections can be computed leading to

ZA(E2aD6b) =

int infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it)

η3(it)ACY

ab [αβ](it) (67)

and the Mobius strip amplitude for the instanton which as we explained mustbe invariant under the orientifold projection yields

ZM(E2aO6) = plusmnint infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it + 1

2

)

η3(it + 1

2

) ACYaa [α

β](it + 1

2

)(68)

3The contribution of the CP-odd Rminus sector is expected to yield corrections to the θ-angle

18

The overall plus sign is for O(1) instantons reflecting the fact that only for thesethe xmicro and θα zero modes survive the orientifold projection Note that up to theargument the Mobius thresholds are ZA(E2aD6a) Therefore for rigid branesthe massless sector reflects the number of four bosonic and two fermionic zeromodes In section 63 we will discuss the number of zero modes if b1(Ξ) gt 0 Allthese stringy threshold corrections are known to be non-holomorphic Thereforeit is not immediately obvious in which sense the expression (65) is meant andhow one can extract the holomorphic superpotential part Y from it

The CFT disc amplitudes in (65) are also not holomorphic but combinenon-holomorphic Kahler potential contributions and holomorphic superpotentialcontributions in the usual way [38 39 40 41]

〈Φab[~x]〉λaλb=

eK

2 YλaΦax1Φx1x2 ΦxN b λbradicKλaaKax1 KxnbKbλb

(69)

=e

K

2 YλabΦab[x]λbradic

Kλaa Kab[x]Kbλb

(610)

Due to the Kaplunovsky-Louis formula (51) the stringy one-loop amplitudes areknown to include the holomorphic Wilsonian part and contributions from wave-function normalisation Applied to the instanton one-loop amplitudes appearingin Z0(E2a) we write

Z0(E2a) = minus8π2 real(f (1)a )minus ba

2ln

(M2

p

micro2

)minus ca

2Ktree (611)

minus ln

(V3gs

)

tree

+sum

b

|IabNb|2

ln[detKab

]tree

where for the brane and instanton configuration in question the coefficients are

ba =sum

b

|IabNb|2

minus 3 ca =sum

b

|IabNb|2

minus 1 (612)

The constant contributions arise from the Mobius amplitude Inserting (69) and(611) in (65) one realises that the Kahler metrics involving an instanton zeromode and a matter field precisely cancel out so that only the matter metricssurvive as required by the general form (64) Moreover the term exp(K2)comes out just right due to the rule that each disc contains precisely two instantonzero modes The holomorphic piece in (64) can therefore be expressed entirelyin terms of other holomorphic quantities like holomorphic Yukawa couplingsthe holomorphic instanton action and the one-loop holomorphic Wilsonian gaugekinetic function on the E2-brane

YΦa1b1ΦaMbM

=sum

conf

signconf exp(minusSE2)tree exp(minusf (1)

a

)

Yλa1bΦa1b1

[~x1] λb1middot middot Yλa1

bΦaLbL[~xL]λbL

(613)

19

This explicitly shows that knowing the tree-level Kahler potentials computingthe matter field correlator in the instanton background up to one-loop level ings is sufficient to deduce the Wilsonian holomorphic instanton generated super-potential Higher order corrections in gs only come from loop corrections to theKahler potentials

63 Instanton corrections to the gauge kinetic functions

So far we have discussed space-time instanton corrections to the superpotentialThese involved one-loop determinants which are given by annulus vacuum di-agrams with at least one E2 boundary These are related to one-loop gaugethreshold corrections to the gauge theory on a D6-brane wrapping the same cycleas the E2 instanton

Now we can ask what other corrections these space-time instantons can induceBy applying S- and T-dualities to the story of world-sheet instanton correctionsin the heterotic string we expect that there can also be E2-instanton corrections

to the holomorphic gauge kinetic functions In the heterotic case similar to thetopological Type II string such corrections arise from string world-sheets of Eulercharacteristic zero ie here from world-sheets with two boundaries Thereforewe expect such corrections to appear for E2-instantons admitting one complexopen string modulus ie those wrapping a three-cycle with Betti number b1(Ξ) =1

Let us start by discussing the instanton zero mode structure for such a cycleFirst let us provide the form of the vertex operators The bosonic fields in the(minus1) ghost picture are

V (minus1)y (z) = y eminusϕ(z)Σh= 1

2q=plusmn1(z) (614)

which before the orientifold projection are accompanied by the two pairs offermionic zero modes

V (minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 1

2(z) (615)

and

V(minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=+ 1

2(z) (616)

Now one has to distinguish two cases depending on how the anti-holomorphicinvolution σ acts on the open string modulus Y

σ y rarr plusmny (617)

In the case that y is invariant under σ called first kind in the following theorientifold projection acts in the same way as for the 4D fields Xmicro ie thetwo bosonic zero modes y and the two fermionic zero modes micro survive In the

20

other case dubbed second kind the bosonic zero mode is projected out andonly the fermionic modulino zero mode micro survives4 Therefore in the absence ofany additional zero modes for instance from E2-D6 intersections the zero modemeasure in any instanton amplitude assumes the following form

intd4x d2θ d2y d2micro eminusSE2 for σ y rarr y (618)

andintd4x d2θ d2micro eminusSE2 for σ y rarr minusy (619)

As an example consider the set-up in figure 3 with σ yi rarr minusyi Here the

x x x2 31

1 2 3y y y

O6

E2

∆x1 ∆ y3

∆x2

Figure 3 Deformations of an instanton which is invariant under the orientifold pro-jection

deformations ∆x12 are of the first kind and ∆y3 is of the second kindNow it is clear that an instanton with precisely one set of fermionic zero

modes of the second kind and no additional zero modes can generate a correctionto the SU(Na) gauge kinetic function The instanton amplitude takes on thefollowing form

〈Fa(p1)Fa(p2)〉E2 =

intd4x d2θ d2micro exp(minusSE2) exp (Z prime

0(E2)) AF 2a(E2 D6a)

where AF 2a(E2 D6a) is the annulus diagram in figure 4 which absorbs all the

appearing fermionic zero modes and where the gauge boson vertex operators inthe (0)-ghost picture have the usual form

V(0)A (z) = ǫmicro (partmicroX(z) + i(p middot ψ)ψmicro(z)) e

ipmiddotX(z) (620)

4By duality this distinction is related to the two kinds of deformations of genus g curvesstudied in [42] The first kind are the curves moving in families ie transversal deformationof the curve The second kind is related to the deformations coming with the genus g of thecurve

21

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

not been explained explicitly in which sense the computed instantonic correlationfunctions are meant to be holomorphic With the results from the first part ofthis paper we clarify this point

Finally we show that E2-instantons not only contribute to the superpotentialbut from the zero mode counting can also contribute to the holomorphic gaugekinetic functions for the SU(N) gauge groups localised on the D6-branes Inorder for such corrections to arise the E2-instanton must not be rigid but mustadmit one extra pair of fermionic zero modes arising from a deformation of theinstanton This is the space-time instanton generalisation of a fact known fromtopological string theory namely that world-sheet instantons induce tr(W 2)hminus1

couplings if they have h boundaries We will see that such couplings can also arisefrom space-time E2-instantons Finally we find that the zero mode counting alsoallows E2-instanton corrections to the FI-terms on the D6-branes Similar to theone-loop corrections these can arise once the supersymmetry on the E2-braneis softly broken by for instance turning on the C3-form modulus through theworld-volume of E2

2 A non-renormalisation theorem

Let us investigate the structure of perturbative and non-perturbative correctionsto the holomorphic gauge kinetic functions for Type II orientifolds We discussthis for Type IIA orientifolds but this is of course related via mirror symmetryto the corresponding Type IIB orientifolds

Consider a Type IIA orientifold with O6-planes and intersecting D6-branespreserving N = 1 supersymmetry in four dimensions ie the D6-branes wrapspecial Lagrangian (sLag) three-cycles Πa of the underlying CalabindashYau manifoldX all preserving the same supersymmetry On the threefold we introduce in theusual way a symplectic basis (AI B

I) I = 0 1 h21 of homological three-cycles with the topological intersection numbers

AI BJ = δJI (21)

Moreover we assume that the AI cycles are invariant under the orientifold pro-jection and that the BJ cycles are projected out The complexified complexstructure moduli on such an orientifold are defined as

U cI =

1

(2π) ℓ3s

[eminusφ4

int

AI

real(Ω3) minus i

int

AI

C3

] (22)

where Ω3 denotes the normalised holomorphic three-form on X and the four-dimensional dilaton is defined by φ4 = φ10 minus 1

2ln(VXℓ

6s) Expanding a three

cycle Πa into the symplectic basis

Πa =M Ia AI +NaI B

I (23)

4

with M I NI isin Z from dimensional reduction of the Dirac-Born-Infeld (DBI)action one can deduce the SU(Na) gauge kinetic functions at string tree-level

fa =

h21sum

I=0

M Ia U

cI (24)

Since the imaginary parts of the U cI are axionic fields they enjoy a Peccei-Quinn

shift symmetry U cI rarr U c

I + cI which is preserved perturbatively and only brokenby E2-brane instantons

Let Ci denote a basis of anti-invariant 2-cycles ie Ci isin H11minus The complex-

ified Kahler moduli are then defined as

T ci =

1

ℓ2s

(int

Ci

J2 minus i

int

Ci

B2

) (25)

where B2 denotes the NS-NS two-form of the Type IIA superstring Thereforealso the complexified Kahler moduli enjoy a Peccei-Quinn shift symmetry brokenby world-sheet instantons Note that the chiral fields T c

i organise the σ-modelperturbation theory and do not contain the dilaton so that the string pertur-bative theory is entirely defined by powers of the U c

I Moreover to shorten thenotation we denote by U c

I and T ci the complexified moduli and by UI and Ti only

the real partsThe superpotentialW and the gauge kinetic function f in the four-dimensional

effective supergravity action are holomorphic quantities In the usual way em-ploying holomorphy and the Peccei-Quinn symmetries above one arrives at thefollowing two non-renormalisation theorems

The superpotential can only have the following dependence on U cI and T c

i

W = Wtree +W np(eminusUc

I eminusT ci

) (26)

ie beyond tree-level there can only be non-perturbative contributions fromworld-sheet and E2-brane instantons Similarly the holomorphic gauge kineticfunction must look like

fa =sum

I

M IaU

cI + f 1-loop

a

(eminusT c

i

)+ fnp

a

(eminusUc

I eminusT ci

) (27)

ie in particular its one-loop correction must not depend on the complex structuremoduli Finally we consider the Fayet-Iliopoulos terms for the U(1)a gaugefields on the D6-branes At string tree-level and for small deviations from thesupersymmetry locus these are given by

ξa = eminusφ4

int

Πa

image(Ω3) = eminusφ4 N Ia

int

BI

image(Ω3) (28)

5

and therefore only depend on the complex structure moduli At this classicallevel there are no αprime corrections It is an important question about brane stabilitywhether these FI-terms receive perturbative or non-perturbative corrections in gsAgain non-renormalisation theorems say that in the Wilsonian sense one expectsperturbative corrections at most at one-loop

In the following we will be concerned with the terms beyond tree-level appear-ing in (26) (27) and for the FI-terms First we discuss the one-loop thresholdcorrections f 1minusloop

(eminusT c

i

) which also make their appearance in the space-time

instanton generated superpotential W np(eminusUc

I eminusT ci

) Second we will revisit the

computation of stringy one-loop corrections to the FI-terms Finally we willdiscuss fnp

(eminusUc

I eminusT ci

)as well as instanton corrections to the FI-terms

3 One-loop thresholds for intersecting D6-branes

on T6

The purpose of this section is to recall the one-loop results for the gauge thresh-old corrections in intersecting D6-brane models [5 30 6] The gauge couplingconstants of the various gauge group factors Ga in such a model up to one loophave the form

8π2

g2a(micro)=

8π2

g2astring+ba2

ln

(M2

s

micro2

)+

∆a

2 (31)

where ba is the beta function coefficient The first term corresponds to the gaugecoupling constant at the string scale which contains the tree-level gauge couplingas well as the ldquouniversalrdquo contributions at one-loop (see section 52) Thesecontributions are universal in the sense that they originate from a redefinition ofthe dilaton and complex structure moduli at one-loop The redefinition is branestack and therefore gauge group independent However as the gauge couplingsdiffer for the various gauge groups already at tree level this correction effectivelyis gauge group dependent The second term gives the usual one-loop running ofthe coupling constants and the third term denotes the one-loop string thresholdcorrections originating from integrating out massive string excitations The lasttwo terms can be computed as a sum of all annulus and Mobius diagrams withone boundary on brane a in the presence of a background magnetic field in thefour-dimensional space-time

ba ln

(M2

s

micro2

)+∆a =

sum

b

TA(D6aD6b) +sum

bprime

TA(D6aD6bprime) (32)

+ TA(D6aD6aprime) + TM(D6aO6)

Here D6cprime denotes the orientifold image of brane c In an orbifold one also hasto take into account the orbifold images of the branes and orientifold planes

6

The relevant amplitudes for the Z2timesZ2 orbifold have been computed [5 6] Ina sector preserving N = 1 supersymmetry (this means in particular

sumI θ

Iab = 0)

the annulus and Mobius amplitudes are (after subtracting terms which uponsumming over all diagrams vanish due to the tadpole cancellation condition) [6]

TA(D6aD6b) =IabNb

2

[ln

(M2

s

micro2

) 3sum

I=1

sign(θIab)minus

minus ln3prod

I=1

(Γ(|θIab|)

Γ(1minus |θIab|)

)sign(θIab)

minus3sum

I=1

sign(θIab) (ln 2minus γ)

] (33)

TM(D6aO6k) = plusmnIaO6k

[ln

(M2

s

micro2

) 3sum

I=1

sign(θIaO6k)minus

minus ln3prod

I=1

(Γ(2|θIaO6k

|)Γ(1minus 2|θIaO6k

|)

)sign(θIaO6k

)

+3sum

I=1

sign(θIaO6k)(γ minus 3 ln 2)

] (34)

where Iab is the intersection number of branes a and b Nb is the number ofbranes on stack b and πθIab is the intersection angle of branes a and b on the Irsquothtorus Similarly IaO6k denotes the intersection number of brane a and orientifoldplane k and πθIaO6k

their intersection angle The formula for TM is only valid for|θIaO6k

| lt 12 the formulas for other cases look similar [6]In a sector preserving N = 2 supersymmetry one finds [5]

TA(D6aD6b) = Nb|IJab IKab |[ln

(M2

s

micro2

)minus ln |η(i T c

I )|4 minus ln(TI VaI ) + γE minus ln(4π)

]

(35)where I denotes the torus on which the branes lie on top of each other TI itsKahler modulus and T c

I its complexification with TI = real(T cI ) Furthermore

V aI = |nI

a + iuImIa|2uI with uI the complex structure modulus of the torus and

nIam

Ia the wrapping numbers on the Irsquoth torus Note that the moduli dependence

of the one-loop threshold function in the N = 2 sectors is in complete agreementwith the non-renormalisation theorems of section two (see eq (27)) since theholomorphic part of ∆N=2

a is proportional only to ln η(i T cI )

For theN = 1 sectors the one-loop thresholds in a given open string D6-branesector have the following form (specialising to the case θ12ab gt 0 θ3ab lt 0)

∆a = minus ba16π2

ln

[Γ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab)

] (36)

This expression is a non-holomorphic function of the complex structure moduliU cI Hence for the N = 1 sectors the holomorphic one-loop gauge kinetic func-

tion f 1minusloopa

(eminusT c

i

)vanishes The emergence of the non-holomorphic terms in the

one-loop threshold corrections will be further discussed in section 5

7

4 Fayet-Iliopoulos terms

In this section we investigate one-loop corrections to the FI-terms for a U(1)agauge field on the D6a-brane induced by the presence of other branes D6b Suchcorrections for Type I string vacua have already been studied in [31 32] and thecase of intersecting D6-branes has been discussed in [7] Here we are followingessentially the computational technique of [7] The crucial observation is thatthe vertex operator for the auxiliary D-field in the (0)-ghost picture is simply

given by the internal world-sheet U(1) current ie V(0)Da

= JU(1) Therefore the

one-point function of V(0)Da

on the annulus with boundaries a and b can be writtenas

〈VDa〉 = minus i

2πpartν

int infin

0

dt Zab(ν it)|ν=0 (41)

where Zab(ν it) denotes the annulus partition function with insertion of exp(2πiJ0)in the open string sector (ab) where J0 is the zero mode of the U(1) current Inthe case of intersecting D6-branes on a torus preserving N = 1 supersymmetryand after application of the Riemann theta-identities this partition function isgiven by

Zab(ν it) = IabNb(minusi)3π4t2

ϑ1(3ν2 it)

prodI ϑ1(minusν

2+ iθI

2t it)

η3(it)prod

I ϑ1(iθI2t it)

(42)

Using that ϑ1(0 it) = 0 and ϑprime1(ν it)|ν=0 = minus2πη3 one obtains the divergence

〈VDa〉 ≃ IabNb

int infin

0

dt

t2 (43)

which is cancelled by tadpole cancellation in a global model Therefore oncethe D6-branes are supersymmetric at tree-level no FI-term is generated at one-loop level and the system is not destabilised This result is consistent with thecomputations in [31 32]

However this is not the end of the story of the one-loop corrections to FI-terms in intersecting D6-brane models One can also envision that a tree-levelFI-term on a brane D6b induces via a one-loop diagram an FI-term on a braneD6a To proceed we assume that θIb rarr θIb + 2ǫI with

sumǫI = ǫ and compute

〈VDa〉ǫI = minus i

2πpartνpartǫI

int infin

0

dt Zab(ǫI ν it)|ν=0ǫI=0 (44)

with

Zab(ǫI ν it) = IabNb

(minusi)3π4t2

ϑ1(3ν2+ i ǫ

2t it)

prodI ϑ1(minusν

2+ i (ǫ

IminusǫJminusǫK)2

t+ iθI2t it)

η3(it)prod

I ϑ1(iǫIt+ iθI

2t it)

8

The derivative with respect to the supersymmetry breaking parameters ǫI bringsdown one factor of t and it turns out that the result is the same for all partǫI

〈VDa〉ǫ ≃ i IabNb

int infin

0

dt

t

3sum

I=1

ϑprime1ϑ1

(iθI

abt

2 it2

) (45)

This is the same expression as the one-loop threshold corrections TA(D6a D6b)so that at linear order in ǫ we obtain for the FI-terms

(ξa)ǫ = (ǫb minus ǫa) TA(D6a D6b) (46)

Completely analogously one can show that this formula remains true also forN = 2 open string sectors Therefore we would like to propose that such arelation between gauge threshold and one-loop corrections to FI-terms is validfor general intersecting D6-brane models Moreover the Wilsonian part of thethresholds TA(D6a D6b) which we compute in the next section should also bethe Wilsonian part of the correction to the FI-terms

5 Wilsonian gauge kinetic function and σ-model

anomalies

In a supersymmetric gauge theory one can compute the running gauge couplingsga(micro

2) in terms of the gauge kinetic functions fa the Kahler potential K and theKahler metrics of the charged matter fields Kab(micro2) [33 8 9 10 11 12 13]

8π2

g2a(micro2)

= 8π2real(fa) +ba2

ln

(Λ2

micro2

)+ca2K

+T (Ga) ln gminus2a (micro2)minus

sum

r

Ta(r) ln detKr(micro2) (51)

with

ba =sum

r

nrTa(r)minus 3 T (Ga) ca =sum

r

nrTa(r)minus T (Ga) (52)

and Ta(r) = Tr(T 2(a)) (T(a) being the generators of the gauge group Ga) In addi-

tion T (Ga) = Ta(adjGa) and nr is the number of multiplets in the representationr of the gauge group and the sums run over these representations In this contextthe natural cutoff scale for a field theory is the Planck scale ie Λ2 =M2

PlThe left hand side of eq (51) is given by eq (31) which contains the gauge

coupling at the string scale 1g2astring as well as the one-loop string thresholdcorrections ∆a In general ∆a is the sum of a non-holomorphic term plus thereal-part of a holomorphic threshold correction

∆a = ∆nha + real(∆hol

a ) (53)

9

On the right hand side of eq (51) fa denotes the Wilsonian ie holomorphicgauge kinetic function which is given in terms of a holomorphic tree-level functionplus the holomorphic part of the one-loop threshold corrections (cf eq (27))

fa = f treea + f 1minusloop

a

(eminusT c

i

)=sum

I

M IaU

cI +∆hol

a (54)

In addition on the right hand side of eq (51) the non-holomorphic terms propor-tional to the Kahler metric of the moduli K and the matter field Kahler metricsKr are due to the one-loop contributions of massless fields These fields generatenon-local terms in the one-loop effective action which correspond to one-loopnon-invariances under σ-model transformations the so-called σ-model anomalies(Kahler and reparametrisation anomalies)

Matching up all terms in eq (51) essentially means that the σ-model anoma-lies can be cancelled in a two-fold way First by local contributions to the gaugecoupling constant via the one-loop threshold contributions ∆a These terms origi-nate from massive string states The second way to cancel the σ-model anomaliesis due to a field dependent (however gauge group independent) one-loop contribu-tion to the Kahler potential of the chiral moduli fields It implies that some of themoduli fields transform non-trivially under the Kahler transformations and alsounder reparametrisations in moduli space The universal one-loop modificationof the Kahler potential is nothing else than a generalised Green-Schwarz mecha-nism cancelling the σ-model anomalies This is analogous to the Green-Schwarzmechanism which cancels anomalies of physical U(1) gauge fields whereas theσ-model anomalies correspond to unphysical composite gauge connections Ef-fectively it means that the Green-Schwarz mechanism with respect to the σ-modelanomalies can be described by a non-holomorphic one-loop field redefinition ofthe associated tree-level moduli fields

As we will see in type IIA orientifold models these field redefinitions act onthe real parts of the dilaton field S as well as the complex structure moduli UJ

S rarr S + δGS(U T )

UJ rarr UJ + δGSJ (U T ) (55)

These redefined fields are those that determine the gauge coupling constants1g2astring at the string scale Recall that as the tree-level gauge couplings (24)are already gauge group dependent so are these one-loop corrections but theonly dependence arises due to the universal one-loop redefinition of the modulifields It is in this sense that we still call these one-loop corrections to the gaugecouplings universal Note that in heterotic string compactifications the σ-modelGreen-Schwarz mechanism only acts on the heterotic dilaton field

10

51 Holomorphic gauge couplings for toroidal models

In summary equation (51) is to be understood recursively which means that onecan insert the tree-level results into the last three terms of eq (51) In additionone also has to include the universal field redefinition eq (55) in 1g2astring inthe left hand side of eq (51) in order to get a complete matching of all termsin eq (51) as we will demonstrate for the aforementioned toroidal orbifold inthe following For N = 2 sectors the one-loop threshold corrections to the gaugecoupling constant indeed contain a holomorphic Wilsonian term f

(1)a whereas

for N = 1 sectors only the non-holomorphic piece ∆nha is present

Specifically the holomorphic gauge kinetic function can now be determinedby comparing the string theoretical formula (31) for the effective gauge couplingwith the field theoretical one (51) The first thing to notice are the differentcutoff scales appearing in the two formulas One needs to convert one into theother using

M2s

M2Pl

prop exp(2φ4) prop (S U1 U2 U3)minus 1

2 (56)

Here φ4 is the four dimensional dilaton and the complex structure moduli in thesupergravity basis can be expressed in terms of φ4 and the complex structuremoduli uI = RI2RI1 as

S =1

2πeminusφ4

1radicu1 u2 u3

UI =1

2πeminusφ4

radicuJ uKuI

with I 6= J 6= K 6= I (57)

These fields are the real parts of complex scalars of four dimensional chiral mul-tiplets Sc and U c

I As N = 4 super YangndashMills theory is finite one expects the sum of the terms

in (31) proportional to T (Ga) to cancel This is because the only chiral multipletstransforming in the adjoint representation of the gauge group are the open stringmoduli which (on the background considered) assemble themselves into threechiral multiplets thus forming an N = 4 sector together with the gauge fieldsTo show that this cancellation does happen one notices the following Firstlynadjoint = 3 as explained such that there is no term in ba proportional to T (Ga)Secondly

K = minus ln(Sc + Sc)minus

3sum

I=1

ln(U cI + U

c

I)minus3sum

I=1

ln(T cI + T

c

I) (58)

gminus2atree = S

3prod

I=1

nIa minus

3sum

I=1

UI nIam

Jam

Ka I 6= J 6= K 6= I (59)

where TI are the Kahler moduli of the torus and nIa m

Ia are the wrapping numbers

of the brane Finally one needs the matter metric for the open string moduli

11

which can be obtained from the T-dual expression in models with D9- and D5-branes [4] Performing the T-duality which essentially amounts to exchangingKahler and complex structure moduli and converting gauge flux into non-trivialintersection angles for the D6-branes one arrives at (I = 1 3)1

KIij =

δijTIUI

∣∣∣∣∣(nJ

a + iuJ mJa )(n

Ka + iuK m

Ka )

(nIa + iuI mI

a)

∣∣∣∣∣ I 6= J 6= K 6= I (510)

Let us now turn to the fields in the fundamental representation of the gaugegroup Ga in particular to the fields arising from the intersection with one otherstack of branes denoted by b For an N = 1 open string sector the metric forthese fields can be written as [34 4] (see also [35])

Kabij = δij S

minusα

3prod

I=1

Uminus(β+ξ θI

ab)

I Tminus(γ+ζ θI

ab)

I

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (511)

where α β γ ξ and ζ are undetermined constants As θ12ab gt 0 and θ3ab lt 0which is assumed in (511) the intersection number Iab is positive implying that

nf = IabNb (512)

Using Ta(f) =12and relations (52 56 58 511 512) one finds a contribution

to the right hand side of (51) proportional to

IabNb

2

(ln

(M2

s

micro2

)+ (2γ minus 1) ln(T1T2T3) + (2β minus 1

2) ln(U1U2U3) + (2αminus 1

2) lnS

+ζ

3sum

I=1

θIab lnTI + ξ

3sum

I=1

θIab lnUI minus ln

[Γ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab)

]) (513)

Using (36) one finds that the first and the last term exactly reproduce the contri-bution of the last two terms in (31) The terms proportional to ζ and ξ will laterbe shown to constitute the aforementioned universal gauge coupling correctionThe remaining three terms can neither be attributed to such a correction nor canthey be written as the real part of a holomorphic function Thus they cannot bethe one-loop correction to the gauge kinetic function and therefore must vanishThis fixes some of the coefficients in the ansatz (511)

α = β =1

4 γ =

1

2 (514)

1An overall factor involving the wrapping numbers was introduced in this expression in orderto achieve full cancellation This can be done as the expressions used are derived only up tooverall constants [34]

12

The same matching of terms appears between the Mobius diagram plus the annu-lus with boundaries on brane a and its orientifold image and the Kahler metricsfor fields in the symmetric and antisymmetric representation Here one has toreplace θIab and IabNb by θIaaprime = 2θIa and IaaprimeNa in (511) and (33) Apart fromthese replacements the Kahler metric for matter in these representations is alsogiven by (511) with the constants α β γ given in (514)

The corrections to the gauge couplings coming fromN = 2 open string sectorswere seen in the previous section to take on quite a different form They containa term

minus ln |η(i T cI )|4 = minus4Re [ln η(i T c

I )] (515)

which can be written as the real part of a holomorphic function This leads oneto conclude that the gauge kinetic function receives one-loop corrections fromthese sectors Inserting the correct prefactor which from the first term in (35)and the corresponding one in (51) can be seen to be proportional to the betafunction coefficient gives

f (1)a = minusNb |IJab IKab |

4π2ln η(i T c

I ) I 6= J 6= K 6= I (516)

where again I denotes the torus in which the branes lie on top of each other andIJKab are the intersection numbers on the other tori

The term minus ln(TI VaI ) in (35) is not the real part of a holomorphic function

Proceeding as before one finds that the Kahler metric for the hypermultiplet (ortwo chiral multiplets) living at an intersection of branes a and b preserving eightsupercharges must be

KIij =

|nIa + iuI m

Ia|

(UJ UK T J TK)12

I 6= J 6= K 6= I (517)

Apart from the factor in the numerator this is in agreement with the form foundby direct calculations [34 4] The appearance of the numerator is however plau-sible as it also appears in the open string moduli metric and the hypermultipletsunder discussion should feel the Irsquoth torus in the same way

52 Universal threshold corrections

In the following the aforementioned ldquouniversalrdquo gauge coupling corrections willbe discussed They also appear in the heterotic [13] and type I [36 37] string andare related to a redefinition of the dilaton at one-loop [9 8] This stems from thefact that the dilaton really lives in a linear multiplet rather than a chiral one

Our general ansatz for the Kahler metrics for the chiral matter in an N = 1sector contains a factor

3prod

J=1

Uminusξ θJ

ab

J Tminusζ θJ

ab

J (518)

13

which according to (51) appears in the one-loop correction to the gauge couplingconstant Neither is this term reproduced in the string one-loop calculation ofthe coupling nor can it be written as a correction to the holomorphic gaugekinetic function Therefore as is familiar from gauge threshold computationsthere remains the possibility that it can be absorbed into a one-loop correctionto the S and UI chiral superfields In the following we require that such a gaugegroup factor independent universal correction is possible and see how this fixesthe parameters in (518)

The first observation is that in order to get something gauge group indepen-dent the factor (518) actually must have the following form

3prod

J=1

Uminusξprime sign(Iab)θ

Jab

J Tminusζprime sign(Iab)θ

Jab

J (519)

with ξprime and ζ prime independent of the brane For the metrics of fields transformingin the symmetric or antisymmetric representation of the gauge group one hasto replace φab = φa minus φb by φaaprime = 2φa and sign(Iab) by sign(Iaaprime minus IaO6) orsign(Iaaprime + IaO6) respectively Then one computes (K prime denotes the factor (518519) appearing in the full Kahler metric K)

sum

r

Ta(r) ln detKprimer = |Iab|Nb

2ln

[3prod

J=1

Uminusξprime sign(Iab) θ

Jab

J Tminusζprime sign(Iab) θ

Jab

J

](520)

+|Iabprime |Nb

2ln

[3prod

J=1

Uminusξprime sign(Iabprime) θ

Jabprime

J Tminusζprime sign(Iabprime ) θ

Jabprime

J

]

+ Na+22

|IaaprimeminusIaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(IaaprimeminusIaO6) θ

Ja

J Tminus2ζprime sign(IaaprimeminusIaO6) θ

Ja

J

]

+ Naminus22

|Iaaprime+IaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(Iaaprime+IaO6) θ

Ja

J Tminus2ζprime sign(Iaaprime+IaO6) θ

Ja

J

]

After a few steps using |Iab| sign(Iab) = Iab and the tadpole cancellation condi-tion this can be brought to the simple form

sum

r

Ta(r) ln detKprimer = minusn1

an2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

minus3sum

J=1

nJam

Ka m

La

[sum

b

NbmJb n

Kb n

Lb

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

J 6= K 6= L 6= J (521)

Therefore these corrections have precisely the form required for them to be iden-tified with the one-loop correction between the linear superfields appearing in

14

string theory and the chiral superfields used in the supergravity description

SL = S minus 1

8π2

sum

b

Nbm1bm

2bm

3b

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

ULJ = UJ +

1

8π2

sum

b

NbmJb n

Kb n

Lb

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

J 6= K 6= L 6= J (522)

In contrast to eq (59) where the tree-level gauge couplings are determined theone-loop gauge couplings at the string scale have to include the redefined fieldsSL and UL

gminus2astring = SL

3prod

I=1

nIa minus

3sum

I=1

ULI n

Iam

Jam

Ka (523)

In contrast to all (to us) known cases studied in the literature for ξprime 6= 0 thefields which are corrected ie the moduli S and UI also appear in the one-loopredefinition Let us propose an argument why such corrections might be expectedto be absent Due to the anomalous U(1) gauge symmetries the chiral superfieldsS and UI participate in the Green-Schwarz mechanism and therefore transformnon-trivially under U(1) gauge transformations This implies that in order tobe gauge invariant the one-loop corrections in (522) proportional to lnUI mustbe extended in the usual way by UI rarr UI + δaGSVa Computing the resultingFI-terms via the supergravity formula ξa2g

2a = partKpartVa|Va=0 gives besides the

tree-level result depending on SL ULI a one-loop contribution proportional to

ξprimesum

b ξ(0)b φJ

b UJ This has an extra dependence on the complex structure moduliUI However for intersecting D6-branes we have seen that the Wilsonian (super-gravity) FI-Terms are proportional to the Wilsonian gauge threshold correctionswhich depend only on the Kahler moduli via instanton corrections (for N = 1sectors they are even vanishing in the setup at hand) This seems to suggest thatthere should better be no UminusθI

I dependence in the matter field Kahler metricsie ξprime = 0

Moreover in analogy to the heterotic string we expect that for the rangeminus1 le θJ le 1 the exponent of TJ runs over the range [minus1 0] This conditionwould fix ζ prime = plusmn12

Let us summarise the conclusions we have drawn from requiring holomorphyof the Wilsonian gauge kinetic function First we provided arguments that theKahler metric for N = 1 chiral matter fields in intersecting D6-brane models isof the following form

Kabij = δij S

minus 14

3prod

J=1

Uminus 1

4J T

minus( 12plusmn 1

2sign(Iab) θ

Jab)

J

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (524)

15

where supersymmetry of course requiressum3

I=1 θIab = 0 Second the holomorphic

gauge kinetic function (on the background considered) only receives correctionsfrom N = 2 open string sectors and the one-loop correction takes on the followingform

f (1)a = minus

sum

b

Nb |IJabIKab |4π2

ln η(i T cI ) I 6= J 6= K 6= I (525)

where the sum only runs over branes b which lie on top of brane a in exactly onetorus denoted by I Therefore the results for the gauge threshold corrections andthe matter field Kahler metrics are consistent both with the non-renormalisationtheorem from section 2 and the Kaplunovsky-Louis formula (51) Clearly itwould be interesting along the lines of [34] to carry out a string amplitude com-putation to fix the free coefficient in the ansatz (511) and see whether our indirectarguments are correct

6 Holomorphic E2-instanton amplitudes

Space-time instantons are given also by D-branes which in this case are EuclideanD2-branes (so-called E2-branes) wrapping three-cycles Ξ in the CalabindashYau sothat they are point-like in four-dimensional Minkowski space Such instantonscan contribute to the holomorphic superpotential and gauge kinetic functions onlyif they preserve half of the N = 1 supersymmetry This means that the instantonmeasure must contain a factor d4x d2θ Let us first clarify an important aspectof this half-BPS condition In the second part of this section we then revisitthe computation of contributions of such instantons to the superpotential andalso clarify some issues concerned with the appearing one-loop determinants Inthe third and fourth part we investigate under which conditions such stringinstantons can also contribute to the gauge kinetic functions and FI-terms

61 Half-BPS instantons

As has been explained in [24 18 20] just wrapping an E2-instanton around arigid sLag three-cycle in the Calabi-Yau gives four bosonic and four fermioniczero modes The vertex operators for the latter are

V(minus12)θ (z) = θα e

minusϕ(z)2 (z)Sα(z) Σh= 3

8q= 3

2(z) (61)

and

V(minus12)

θ(z) = θα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 3

2(z) (62)

Therefore if the instanton is not invariant under the orientifold projection onestill has four instead of the desired two fermionic zero modes Thus only by

16

placing the E2-brane in a position invariant under Ωσ does one have a chanceto get rid of the two additional zero modes θ For so called O(n) instantons onecan see that the zero modes xmicro θ are symmetrised and the mode θ gets anti-symmetrised For the opposite projection ie for USp(2n) instantons the zeromodes xmicro θ are anti-symmetrised and the mode θ gets symmetrised Thereforeone can only get the simple d4x d2θ instanton measure for a single O(1) instanton

62 Superpotential contributions

In order to contribute to the superpotential we also require that there do notarise any further zero modes from E2-E2 open strings so that the three-cycleΞ should be rigid ie b1(Ξ) = 0 Therefore considering an E2-instanton in anintersecting brane configuration additional zero modes can only arise from theintersection of the instanton Ξ with D6-branes Πa There are Na [ΞcapΠa]

+ chiralfermionic zero modes λaI and Na [Ξ cap Πa]

minus anti-chiral ones λaJ 2

For its presentation it is useful to introduce the short-hand notation

Φak bk [~xk] = Φak xk1middot Φxk1xk2

middot Φxk2xk3middot middot Φxknminus1xkn

middot Φxkn(k)bk (63)

for the chain-product of open string vertex operators Here we define Φak bk [~0] =Φak bk

To extract the superpotential one can probe it by evaluating an appropriatematter field correlator in the instanton background The CFT allows one tocompute it in physical normalisation which combines the superpotential part Ywith the matter field Kahler metrics like

〈Φa1b1 middot middot ΦaM bM 〉E2minusinst =e

K

2 YΦa1b1ΦaMbMradic

Ka1b1 middot middotKaM bM

(64)

In [14] a general expression for the single E2-instanton contribution to thecharged matter superpotential was proposed involving the evaluation of the fol-lowing zero mode integral over disc and one-loop open string CFT amplitudes

〈Φa1b1 middot middot ΦaM bM 〉E2 =V3gs

intd4x d2θ

sum

conf

proda

(prod[ΞcapΠa]+

i=1 dλia) (prod[ΞcapΠa]minus

i=1 dλi

a

)

exp(minusSE2) exp (Z prime0(E2)) 〈Φa1b1[~x1]〉λa1 λb1

middot middot 〈ΦaLbL[~xL]〉λaLλbL

(65)

For simplicity we do not consider the case that matter fields are also assignedto string loop diagrams The one-loop contributions are annulus diagrams for

2Here we introduced the physical intersection number between two branes Πa cap Πb whichis the sum of positive [Πa capΠb]

+ and negative [Πa cap Πb]minus intersections

17

open strings with one boundary on the E2-instanton and the other boundary onthe various D6-branes and Mobius diagrams with boundary on the E2-instanton

〈1〉1-loop = Z prime0(E2) =

sumb Z

primeA(E2aD6b) + Z primeM(E2aO6) (66)

Here Z prime means that we only sum over the massive open string states in theloop amplitude as the zero modes are taken care of explicitly It was shownthat these instantonic open string loop diagrams are identical to the one-loopthreshold corrections TA(D6aD6b) Diagrammatically we have the intriguingrelation shown in figure 1 and in figure 2 which holds for the even spin structures3

Fa

Fa x

x

D6aaE2 =D6b D6b

Figure 1 Relation between instantonic one-loop amplitudes and corresponding gaugethreshold corrections

Fa

Fa x

x

D6aaE2 =

O6O6

Figure 2 Relation between instantonic Mobius amplitude and corresponding gaugethreshold corrections

The annulus threshold corrections can be computed leading to

ZA(E2aD6b) =

int infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it)

η3(it)ACY

ab [αβ](it) (67)

and the Mobius strip amplitude for the instanton which as we explained mustbe invariant under the orientifold projection yields

ZM(E2aO6) = plusmnint infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it + 1

2

)

η3(it + 1

2

) ACYaa [α

β](it + 1

2

)(68)

3The contribution of the CP-odd Rminus sector is expected to yield corrections to the θ-angle

18

The overall plus sign is for O(1) instantons reflecting the fact that only for thesethe xmicro and θα zero modes survive the orientifold projection Note that up to theargument the Mobius thresholds are ZA(E2aD6a) Therefore for rigid branesthe massless sector reflects the number of four bosonic and two fermionic zeromodes In section 63 we will discuss the number of zero modes if b1(Ξ) gt 0 Allthese stringy threshold corrections are known to be non-holomorphic Thereforeit is not immediately obvious in which sense the expression (65) is meant andhow one can extract the holomorphic superpotential part Y from it

The CFT disc amplitudes in (65) are also not holomorphic but combinenon-holomorphic Kahler potential contributions and holomorphic superpotentialcontributions in the usual way [38 39 40 41]

〈Φab[~x]〉λaλb=

eK

2 YλaΦax1Φx1x2 ΦxN b λbradicKλaaKax1 KxnbKbλb

(69)

=e

K

2 YλabΦab[x]λbradic

Kλaa Kab[x]Kbλb

(610)

Due to the Kaplunovsky-Louis formula (51) the stringy one-loop amplitudes areknown to include the holomorphic Wilsonian part and contributions from wave-function normalisation Applied to the instanton one-loop amplitudes appearingin Z0(E2a) we write

Z0(E2a) = minus8π2 real(f (1)a )minus ba

2ln

(M2

p

micro2

)minus ca

2Ktree (611)

minus ln

(V3gs

)

tree

+sum

b

|IabNb|2

ln[detKab

]tree

where for the brane and instanton configuration in question the coefficients are

ba =sum

b

|IabNb|2

minus 3 ca =sum

b

|IabNb|2

minus 1 (612)

The constant contributions arise from the Mobius amplitude Inserting (69) and(611) in (65) one realises that the Kahler metrics involving an instanton zeromode and a matter field precisely cancel out so that only the matter metricssurvive as required by the general form (64) Moreover the term exp(K2)comes out just right due to the rule that each disc contains precisely two instantonzero modes The holomorphic piece in (64) can therefore be expressed entirelyin terms of other holomorphic quantities like holomorphic Yukawa couplingsthe holomorphic instanton action and the one-loop holomorphic Wilsonian gaugekinetic function on the E2-brane

YΦa1b1ΦaMbM

=sum

conf

signconf exp(minusSE2)tree exp(minusf (1)

a

)

Yλa1bΦa1b1

[~x1] λb1middot middot Yλa1

bΦaLbL[~xL]λbL

(613)

19

This explicitly shows that knowing the tree-level Kahler potentials computingthe matter field correlator in the instanton background up to one-loop level ings is sufficient to deduce the Wilsonian holomorphic instanton generated super-potential Higher order corrections in gs only come from loop corrections to theKahler potentials

63 Instanton corrections to the gauge kinetic functions

So far we have discussed space-time instanton corrections to the superpotentialThese involved one-loop determinants which are given by annulus vacuum di-agrams with at least one E2 boundary These are related to one-loop gaugethreshold corrections to the gauge theory on a D6-brane wrapping the same cycleas the E2 instanton

Now we can ask what other corrections these space-time instantons can induceBy applying S- and T-dualities to the story of world-sheet instanton correctionsin the heterotic string we expect that there can also be E2-instanton corrections

to the holomorphic gauge kinetic functions In the heterotic case similar to thetopological Type II string such corrections arise from string world-sheets of Eulercharacteristic zero ie here from world-sheets with two boundaries Thereforewe expect such corrections to appear for E2-instantons admitting one complexopen string modulus ie those wrapping a three-cycle with Betti number b1(Ξ) =1

Let us start by discussing the instanton zero mode structure for such a cycleFirst let us provide the form of the vertex operators The bosonic fields in the(minus1) ghost picture are

V (minus1)y (z) = y eminusϕ(z)Σh= 1

2q=plusmn1(z) (614)

which before the orientifold projection are accompanied by the two pairs offermionic zero modes

V (minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 1

2(z) (615)

and

V(minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=+ 1

2(z) (616)

Now one has to distinguish two cases depending on how the anti-holomorphicinvolution σ acts on the open string modulus Y

σ y rarr plusmny (617)

In the case that y is invariant under σ called first kind in the following theorientifold projection acts in the same way as for the 4D fields Xmicro ie thetwo bosonic zero modes y and the two fermionic zero modes micro survive In the

20

other case dubbed second kind the bosonic zero mode is projected out andonly the fermionic modulino zero mode micro survives4 Therefore in the absence ofany additional zero modes for instance from E2-D6 intersections the zero modemeasure in any instanton amplitude assumes the following form

intd4x d2θ d2y d2micro eminusSE2 for σ y rarr y (618)

andintd4x d2θ d2micro eminusSE2 for σ y rarr minusy (619)

As an example consider the set-up in figure 3 with σ yi rarr minusyi Here the

x x x2 31

1 2 3y y y

O6

E2

∆x1 ∆ y3

∆x2

Figure 3 Deformations of an instanton which is invariant under the orientifold pro-jection

deformations ∆x12 are of the first kind and ∆y3 is of the second kindNow it is clear that an instanton with precisely one set of fermionic zero

modes of the second kind and no additional zero modes can generate a correctionto the SU(Na) gauge kinetic function The instanton amplitude takes on thefollowing form

〈Fa(p1)Fa(p2)〉E2 =

intd4x d2θ d2micro exp(minusSE2) exp (Z prime

0(E2)) AF 2a(E2 D6a)

where AF 2a(E2 D6a) is the annulus diagram in figure 4 which absorbs all the

appearing fermionic zero modes and where the gauge boson vertex operators inthe (0)-ghost picture have the usual form

V(0)A (z) = ǫmicro (partmicroX(z) + i(p middot ψ)ψmicro(z)) e

ipmiddotX(z) (620)

4By duality this distinction is related to the two kinds of deformations of genus g curvesstudied in [42] The first kind are the curves moving in families ie transversal deformationof the curve The second kind is related to the deformations coming with the genus g of thecurve

21

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

with M I NI isin Z from dimensional reduction of the Dirac-Born-Infeld (DBI)action one can deduce the SU(Na) gauge kinetic functions at string tree-level

fa =

h21sum

I=0

M Ia U

cI (24)

Since the imaginary parts of the U cI are axionic fields they enjoy a Peccei-Quinn

shift symmetry U cI rarr U c

I + cI which is preserved perturbatively and only brokenby E2-brane instantons

Let Ci denote a basis of anti-invariant 2-cycles ie Ci isin H11minus The complex-

ified Kahler moduli are then defined as

T ci =

1

ℓ2s

(int

Ci

J2 minus i

int

Ci

B2

) (25)

where B2 denotes the NS-NS two-form of the Type IIA superstring Thereforealso the complexified Kahler moduli enjoy a Peccei-Quinn shift symmetry brokenby world-sheet instantons Note that the chiral fields T c

i organise the σ-modelperturbation theory and do not contain the dilaton so that the string pertur-bative theory is entirely defined by powers of the U c

I Moreover to shorten thenotation we denote by U c

I and T ci the complexified moduli and by UI and Ti only

the real partsThe superpotentialW and the gauge kinetic function f in the four-dimensional

effective supergravity action are holomorphic quantities In the usual way em-ploying holomorphy and the Peccei-Quinn symmetries above one arrives at thefollowing two non-renormalisation theorems

The superpotential can only have the following dependence on U cI and T c

i

W = Wtree +W np(eminusUc

I eminusT ci

) (26)

ie beyond tree-level there can only be non-perturbative contributions fromworld-sheet and E2-brane instantons Similarly the holomorphic gauge kineticfunction must look like

fa =sum

I

M IaU

cI + f 1-loop

a

(eminusT c

i

)+ fnp

a

(eminusUc

I eminusT ci

) (27)

ie in particular its one-loop correction must not depend on the complex structuremoduli Finally we consider the Fayet-Iliopoulos terms for the U(1)a gaugefields on the D6-branes At string tree-level and for small deviations from thesupersymmetry locus these are given by

ξa = eminusφ4

int

Πa

image(Ω3) = eminusφ4 N Ia

int

BI

image(Ω3) (28)

5

and therefore only depend on the complex structure moduli At this classicallevel there are no αprime corrections It is an important question about brane stabilitywhether these FI-terms receive perturbative or non-perturbative corrections in gsAgain non-renormalisation theorems say that in the Wilsonian sense one expectsperturbative corrections at most at one-loop

In the following we will be concerned with the terms beyond tree-level appear-ing in (26) (27) and for the FI-terms First we discuss the one-loop thresholdcorrections f 1minusloop

(eminusT c

i

) which also make their appearance in the space-time

instanton generated superpotential W np(eminusUc

I eminusT ci

) Second we will revisit the

computation of stringy one-loop corrections to the FI-terms Finally we willdiscuss fnp

(eminusUc

I eminusT ci

)as well as instanton corrections to the FI-terms

3 One-loop thresholds for intersecting D6-branes

on T6

The purpose of this section is to recall the one-loop results for the gauge thresh-old corrections in intersecting D6-brane models [5 30 6] The gauge couplingconstants of the various gauge group factors Ga in such a model up to one loophave the form

8π2

g2a(micro)=

8π2

g2astring+ba2

ln

(M2

s

micro2

)+

∆a

2 (31)

where ba is the beta function coefficient The first term corresponds to the gaugecoupling constant at the string scale which contains the tree-level gauge couplingas well as the ldquouniversalrdquo contributions at one-loop (see section 52) Thesecontributions are universal in the sense that they originate from a redefinition ofthe dilaton and complex structure moduli at one-loop The redefinition is branestack and therefore gauge group independent However as the gauge couplingsdiffer for the various gauge groups already at tree level this correction effectivelyis gauge group dependent The second term gives the usual one-loop running ofthe coupling constants and the third term denotes the one-loop string thresholdcorrections originating from integrating out massive string excitations The lasttwo terms can be computed as a sum of all annulus and Mobius diagrams withone boundary on brane a in the presence of a background magnetic field in thefour-dimensional space-time

ba ln

(M2

s

micro2

)+∆a =

sum

b

TA(D6aD6b) +sum

bprime

TA(D6aD6bprime) (32)

+ TA(D6aD6aprime) + TM(D6aO6)

Here D6cprime denotes the orientifold image of brane c In an orbifold one also hasto take into account the orbifold images of the branes and orientifold planes

6

The relevant amplitudes for the Z2timesZ2 orbifold have been computed [5 6] Ina sector preserving N = 1 supersymmetry (this means in particular

sumI θ

Iab = 0)

the annulus and Mobius amplitudes are (after subtracting terms which uponsumming over all diagrams vanish due to the tadpole cancellation condition) [6]

TA(D6aD6b) =IabNb

2

[ln

(M2

s

micro2

) 3sum

I=1

sign(θIab)minus

minus ln3prod

I=1

(Γ(|θIab|)

Γ(1minus |θIab|)

)sign(θIab)

minus3sum

I=1

sign(θIab) (ln 2minus γ)

] (33)

TM(D6aO6k) = plusmnIaO6k

[ln

(M2

s

micro2

) 3sum

I=1

sign(θIaO6k)minus

minus ln3prod

I=1

(Γ(2|θIaO6k

|)Γ(1minus 2|θIaO6k

|)

)sign(θIaO6k

)

+3sum

I=1

sign(θIaO6k)(γ minus 3 ln 2)

] (34)

where Iab is the intersection number of branes a and b Nb is the number ofbranes on stack b and πθIab is the intersection angle of branes a and b on the Irsquothtorus Similarly IaO6k denotes the intersection number of brane a and orientifoldplane k and πθIaO6k

their intersection angle The formula for TM is only valid for|θIaO6k

| lt 12 the formulas for other cases look similar [6]In a sector preserving N = 2 supersymmetry one finds [5]

TA(D6aD6b) = Nb|IJab IKab |[ln

(M2

s

micro2

)minus ln |η(i T c

I )|4 minus ln(TI VaI ) + γE minus ln(4π)

]

(35)where I denotes the torus on which the branes lie on top of each other TI itsKahler modulus and T c

I its complexification with TI = real(T cI ) Furthermore

V aI = |nI

a + iuImIa|2uI with uI the complex structure modulus of the torus and

nIam

Ia the wrapping numbers on the Irsquoth torus Note that the moduli dependence

of the one-loop threshold function in the N = 2 sectors is in complete agreementwith the non-renormalisation theorems of section two (see eq (27)) since theholomorphic part of ∆N=2

a is proportional only to ln η(i T cI )

For theN = 1 sectors the one-loop thresholds in a given open string D6-branesector have the following form (specialising to the case θ12ab gt 0 θ3ab lt 0)

∆a = minus ba16π2

ln

[Γ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab)

] (36)

This expression is a non-holomorphic function of the complex structure moduliU cI Hence for the N = 1 sectors the holomorphic one-loop gauge kinetic func-

tion f 1minusloopa

(eminusT c

i

)vanishes The emergence of the non-holomorphic terms in the

one-loop threshold corrections will be further discussed in section 5

7

4 Fayet-Iliopoulos terms

In this section we investigate one-loop corrections to the FI-terms for a U(1)agauge field on the D6a-brane induced by the presence of other branes D6b Suchcorrections for Type I string vacua have already been studied in [31 32] and thecase of intersecting D6-branes has been discussed in [7] Here we are followingessentially the computational technique of [7] The crucial observation is thatthe vertex operator for the auxiliary D-field in the (0)-ghost picture is simply

given by the internal world-sheet U(1) current ie V(0)Da

= JU(1) Therefore the

one-point function of V(0)Da

on the annulus with boundaries a and b can be writtenas

〈VDa〉 = minus i

2πpartν

int infin

0

dt Zab(ν it)|ν=0 (41)

where Zab(ν it) denotes the annulus partition function with insertion of exp(2πiJ0)in the open string sector (ab) where J0 is the zero mode of the U(1) current Inthe case of intersecting D6-branes on a torus preserving N = 1 supersymmetryand after application of the Riemann theta-identities this partition function isgiven by

Zab(ν it) = IabNb(minusi)3π4t2

ϑ1(3ν2 it)

prodI ϑ1(minusν

2+ iθI

2t it)

η3(it)prod

I ϑ1(iθI2t it)

(42)

Using that ϑ1(0 it) = 0 and ϑprime1(ν it)|ν=0 = minus2πη3 one obtains the divergence

〈VDa〉 ≃ IabNb

int infin

0

dt

t2 (43)

which is cancelled by tadpole cancellation in a global model Therefore oncethe D6-branes are supersymmetric at tree-level no FI-term is generated at one-loop level and the system is not destabilised This result is consistent with thecomputations in [31 32]

However this is not the end of the story of the one-loop corrections to FI-terms in intersecting D6-brane models One can also envision that a tree-levelFI-term on a brane D6b induces via a one-loop diagram an FI-term on a braneD6a To proceed we assume that θIb rarr θIb + 2ǫI with

sumǫI = ǫ and compute

〈VDa〉ǫI = minus i

2πpartνpartǫI

int infin

0

dt Zab(ǫI ν it)|ν=0ǫI=0 (44)

with

Zab(ǫI ν it) = IabNb

(minusi)3π4t2

ϑ1(3ν2+ i ǫ

2t it)

prodI ϑ1(minusν

2+ i (ǫ

IminusǫJminusǫK)2

t+ iθI2t it)

η3(it)prod

I ϑ1(iǫIt+ iθI

2t it)

8

The derivative with respect to the supersymmetry breaking parameters ǫI bringsdown one factor of t and it turns out that the result is the same for all partǫI

〈VDa〉ǫ ≃ i IabNb

int infin

0

dt

t

3sum

I=1

ϑprime1ϑ1

(iθI

abt

2 it2

) (45)

This is the same expression as the one-loop threshold corrections TA(D6a D6b)so that at linear order in ǫ we obtain for the FI-terms

(ξa)ǫ = (ǫb minus ǫa) TA(D6a D6b) (46)

Completely analogously one can show that this formula remains true also forN = 2 open string sectors Therefore we would like to propose that such arelation between gauge threshold and one-loop corrections to FI-terms is validfor general intersecting D6-brane models Moreover the Wilsonian part of thethresholds TA(D6a D6b) which we compute in the next section should also bethe Wilsonian part of the correction to the FI-terms

5 Wilsonian gauge kinetic function and σ-model

anomalies

In a supersymmetric gauge theory one can compute the running gauge couplingsga(micro

2) in terms of the gauge kinetic functions fa the Kahler potential K and theKahler metrics of the charged matter fields Kab(micro2) [33 8 9 10 11 12 13]

8π2

g2a(micro2)

= 8π2real(fa) +ba2

ln

(Λ2

micro2

)+ca2K

+T (Ga) ln gminus2a (micro2)minus

sum

r

Ta(r) ln detKr(micro2) (51)

with

ba =sum

r

nrTa(r)minus 3 T (Ga) ca =sum

r

nrTa(r)minus T (Ga) (52)

and Ta(r) = Tr(T 2(a)) (T(a) being the generators of the gauge group Ga) In addi-

tion T (Ga) = Ta(adjGa) and nr is the number of multiplets in the representationr of the gauge group and the sums run over these representations In this contextthe natural cutoff scale for a field theory is the Planck scale ie Λ2 =M2

PlThe left hand side of eq (51) is given by eq (31) which contains the gauge

coupling at the string scale 1g2astring as well as the one-loop string thresholdcorrections ∆a In general ∆a is the sum of a non-holomorphic term plus thereal-part of a holomorphic threshold correction

∆a = ∆nha + real(∆hol

a ) (53)

9

On the right hand side of eq (51) fa denotes the Wilsonian ie holomorphicgauge kinetic function which is given in terms of a holomorphic tree-level functionplus the holomorphic part of the one-loop threshold corrections (cf eq (27))

fa = f treea + f 1minusloop

a

(eminusT c

i

)=sum

I

M IaU

cI +∆hol

a (54)

In addition on the right hand side of eq (51) the non-holomorphic terms propor-tional to the Kahler metric of the moduli K and the matter field Kahler metricsKr are due to the one-loop contributions of massless fields These fields generatenon-local terms in the one-loop effective action which correspond to one-loopnon-invariances under σ-model transformations the so-called σ-model anomalies(Kahler and reparametrisation anomalies)

Matching up all terms in eq (51) essentially means that the σ-model anoma-lies can be cancelled in a two-fold way First by local contributions to the gaugecoupling constant via the one-loop threshold contributions ∆a These terms origi-nate from massive string states The second way to cancel the σ-model anomaliesis due to a field dependent (however gauge group independent) one-loop contribu-tion to the Kahler potential of the chiral moduli fields It implies that some of themoduli fields transform non-trivially under the Kahler transformations and alsounder reparametrisations in moduli space The universal one-loop modificationof the Kahler potential is nothing else than a generalised Green-Schwarz mecha-nism cancelling the σ-model anomalies This is analogous to the Green-Schwarzmechanism which cancels anomalies of physical U(1) gauge fields whereas theσ-model anomalies correspond to unphysical composite gauge connections Ef-fectively it means that the Green-Schwarz mechanism with respect to the σ-modelanomalies can be described by a non-holomorphic one-loop field redefinition ofthe associated tree-level moduli fields

As we will see in type IIA orientifold models these field redefinitions act onthe real parts of the dilaton field S as well as the complex structure moduli UJ

S rarr S + δGS(U T )

UJ rarr UJ + δGSJ (U T ) (55)

These redefined fields are those that determine the gauge coupling constants1g2astring at the string scale Recall that as the tree-level gauge couplings (24)are already gauge group dependent so are these one-loop corrections but theonly dependence arises due to the universal one-loop redefinition of the modulifields It is in this sense that we still call these one-loop corrections to the gaugecouplings universal Note that in heterotic string compactifications the σ-modelGreen-Schwarz mechanism only acts on the heterotic dilaton field

10

51 Holomorphic gauge couplings for toroidal models

In summary equation (51) is to be understood recursively which means that onecan insert the tree-level results into the last three terms of eq (51) In additionone also has to include the universal field redefinition eq (55) in 1g2astring inthe left hand side of eq (51) in order to get a complete matching of all termsin eq (51) as we will demonstrate for the aforementioned toroidal orbifold inthe following For N = 2 sectors the one-loop threshold corrections to the gaugecoupling constant indeed contain a holomorphic Wilsonian term f

(1)a whereas

for N = 1 sectors only the non-holomorphic piece ∆nha is present

Specifically the holomorphic gauge kinetic function can now be determinedby comparing the string theoretical formula (31) for the effective gauge couplingwith the field theoretical one (51) The first thing to notice are the differentcutoff scales appearing in the two formulas One needs to convert one into theother using

M2s

M2Pl

prop exp(2φ4) prop (S U1 U2 U3)minus 1

2 (56)

Here φ4 is the four dimensional dilaton and the complex structure moduli in thesupergravity basis can be expressed in terms of φ4 and the complex structuremoduli uI = RI2RI1 as

S =1

2πeminusφ4

1radicu1 u2 u3

UI =1

2πeminusφ4

radicuJ uKuI

with I 6= J 6= K 6= I (57)

These fields are the real parts of complex scalars of four dimensional chiral mul-tiplets Sc and U c

I As N = 4 super YangndashMills theory is finite one expects the sum of the terms

in (31) proportional to T (Ga) to cancel This is because the only chiral multipletstransforming in the adjoint representation of the gauge group are the open stringmoduli which (on the background considered) assemble themselves into threechiral multiplets thus forming an N = 4 sector together with the gauge fieldsTo show that this cancellation does happen one notices the following Firstlynadjoint = 3 as explained such that there is no term in ba proportional to T (Ga)Secondly

K = minus ln(Sc + Sc)minus

3sum

I=1

ln(U cI + U

c

I)minus3sum

I=1

ln(T cI + T

c

I) (58)

gminus2atree = S

3prod

I=1

nIa minus

3sum

I=1

UI nIam

Jam

Ka I 6= J 6= K 6= I (59)

where TI are the Kahler moduli of the torus and nIa m

Ia are the wrapping numbers

of the brane Finally one needs the matter metric for the open string moduli

11

which can be obtained from the T-dual expression in models with D9- and D5-branes [4] Performing the T-duality which essentially amounts to exchangingKahler and complex structure moduli and converting gauge flux into non-trivialintersection angles for the D6-branes one arrives at (I = 1 3)1

KIij =

δijTIUI

∣∣∣∣∣(nJ

a + iuJ mJa )(n

Ka + iuK m

Ka )

(nIa + iuI mI

a)

∣∣∣∣∣ I 6= J 6= K 6= I (510)

Let us now turn to the fields in the fundamental representation of the gaugegroup Ga in particular to the fields arising from the intersection with one otherstack of branes denoted by b For an N = 1 open string sector the metric forthese fields can be written as [34 4] (see also [35])

Kabij = δij S

minusα

3prod

I=1

Uminus(β+ξ θI

ab)

I Tminus(γ+ζ θI

ab)

I

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (511)

where α β γ ξ and ζ are undetermined constants As θ12ab gt 0 and θ3ab lt 0which is assumed in (511) the intersection number Iab is positive implying that

nf = IabNb (512)

Using Ta(f) =12and relations (52 56 58 511 512) one finds a contribution

to the right hand side of (51) proportional to

IabNb

2

(ln

(M2

s

micro2

)+ (2γ minus 1) ln(T1T2T3) + (2β minus 1

2) ln(U1U2U3) + (2αminus 1

2) lnS

+ζ

3sum

I=1

θIab lnTI + ξ

3sum

I=1

θIab lnUI minus ln

[Γ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab)

]) (513)

Using (36) one finds that the first and the last term exactly reproduce the contri-bution of the last two terms in (31) The terms proportional to ζ and ξ will laterbe shown to constitute the aforementioned universal gauge coupling correctionThe remaining three terms can neither be attributed to such a correction nor canthey be written as the real part of a holomorphic function Thus they cannot bethe one-loop correction to the gauge kinetic function and therefore must vanishThis fixes some of the coefficients in the ansatz (511)

α = β =1

4 γ =

1

2 (514)

1An overall factor involving the wrapping numbers was introduced in this expression in orderto achieve full cancellation This can be done as the expressions used are derived only up tooverall constants [34]

12

The same matching of terms appears between the Mobius diagram plus the annu-lus with boundaries on brane a and its orientifold image and the Kahler metricsfor fields in the symmetric and antisymmetric representation Here one has toreplace θIab and IabNb by θIaaprime = 2θIa and IaaprimeNa in (511) and (33) Apart fromthese replacements the Kahler metric for matter in these representations is alsogiven by (511) with the constants α β γ given in (514)

The corrections to the gauge couplings coming fromN = 2 open string sectorswere seen in the previous section to take on quite a different form They containa term

minus ln |η(i T cI )|4 = minus4Re [ln η(i T c

I )] (515)

which can be written as the real part of a holomorphic function This leads oneto conclude that the gauge kinetic function receives one-loop corrections fromthese sectors Inserting the correct prefactor which from the first term in (35)and the corresponding one in (51) can be seen to be proportional to the betafunction coefficient gives

f (1)a = minusNb |IJab IKab |

4π2ln η(i T c

I ) I 6= J 6= K 6= I (516)

where again I denotes the torus in which the branes lie on top of each other andIJKab are the intersection numbers on the other tori

The term minus ln(TI VaI ) in (35) is not the real part of a holomorphic function

Proceeding as before one finds that the Kahler metric for the hypermultiplet (ortwo chiral multiplets) living at an intersection of branes a and b preserving eightsupercharges must be

KIij =

|nIa + iuI m

Ia|

(UJ UK T J TK)12

I 6= J 6= K 6= I (517)

Apart from the factor in the numerator this is in agreement with the form foundby direct calculations [34 4] The appearance of the numerator is however plau-sible as it also appears in the open string moduli metric and the hypermultipletsunder discussion should feel the Irsquoth torus in the same way

52 Universal threshold corrections

In the following the aforementioned ldquouniversalrdquo gauge coupling corrections willbe discussed They also appear in the heterotic [13] and type I [36 37] string andare related to a redefinition of the dilaton at one-loop [9 8] This stems from thefact that the dilaton really lives in a linear multiplet rather than a chiral one

Our general ansatz for the Kahler metrics for the chiral matter in an N = 1sector contains a factor

3prod

J=1

Uminusξ θJ

ab

J Tminusζ θJ

ab

J (518)

13

which according to (51) appears in the one-loop correction to the gauge couplingconstant Neither is this term reproduced in the string one-loop calculation ofthe coupling nor can it be written as a correction to the holomorphic gaugekinetic function Therefore as is familiar from gauge threshold computationsthere remains the possibility that it can be absorbed into a one-loop correctionto the S and UI chiral superfields In the following we require that such a gaugegroup factor independent universal correction is possible and see how this fixesthe parameters in (518)

The first observation is that in order to get something gauge group indepen-dent the factor (518) actually must have the following form

3prod

J=1

Uminusξprime sign(Iab)θ

Jab

J Tminusζprime sign(Iab)θ

Jab

J (519)

with ξprime and ζ prime independent of the brane For the metrics of fields transformingin the symmetric or antisymmetric representation of the gauge group one hasto replace φab = φa minus φb by φaaprime = 2φa and sign(Iab) by sign(Iaaprime minus IaO6) orsign(Iaaprime + IaO6) respectively Then one computes (K prime denotes the factor (518519) appearing in the full Kahler metric K)

sum

r

Ta(r) ln detKprimer = |Iab|Nb

2ln

[3prod

J=1

Uminusξprime sign(Iab) θ

Jab

J Tminusζprime sign(Iab) θ

Jab

J

](520)

+|Iabprime |Nb

2ln

[3prod

J=1

Uminusξprime sign(Iabprime) θ

Jabprime

J Tminusζprime sign(Iabprime ) θ

Jabprime

J

]

+ Na+22

|IaaprimeminusIaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(IaaprimeminusIaO6) θ

Ja

J Tminus2ζprime sign(IaaprimeminusIaO6) θ

Ja

J

]

+ Naminus22

|Iaaprime+IaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(Iaaprime+IaO6) θ

Ja

J Tminus2ζprime sign(Iaaprime+IaO6) θ

Ja

J

]

After a few steps using |Iab| sign(Iab) = Iab and the tadpole cancellation condi-tion this can be brought to the simple form

sum

r

Ta(r) ln detKprimer = minusn1

an2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

minus3sum

J=1

nJam

Ka m

La

[sum

b

NbmJb n

Kb n

Lb

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

J 6= K 6= L 6= J (521)

Therefore these corrections have precisely the form required for them to be iden-tified with the one-loop correction between the linear superfields appearing in

14

string theory and the chiral superfields used in the supergravity description

SL = S minus 1

8π2

sum

b

Nbm1bm

2bm

3b

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

ULJ = UJ +

1

8π2

sum

b

NbmJb n

Kb n

Lb

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

J 6= K 6= L 6= J (522)

In contrast to eq (59) where the tree-level gauge couplings are determined theone-loop gauge couplings at the string scale have to include the redefined fieldsSL and UL

gminus2astring = SL

3prod

I=1

nIa minus

3sum

I=1

ULI n

Iam

Jam

Ka (523)

In contrast to all (to us) known cases studied in the literature for ξprime 6= 0 thefields which are corrected ie the moduli S and UI also appear in the one-loopredefinition Let us propose an argument why such corrections might be expectedto be absent Due to the anomalous U(1) gauge symmetries the chiral superfieldsS and UI participate in the Green-Schwarz mechanism and therefore transformnon-trivially under U(1) gauge transformations This implies that in order tobe gauge invariant the one-loop corrections in (522) proportional to lnUI mustbe extended in the usual way by UI rarr UI + δaGSVa Computing the resultingFI-terms via the supergravity formula ξa2g

2a = partKpartVa|Va=0 gives besides the

tree-level result depending on SL ULI a one-loop contribution proportional to

ξprimesum

b ξ(0)b φJ

b UJ This has an extra dependence on the complex structure moduliUI However for intersecting D6-branes we have seen that the Wilsonian (super-gravity) FI-Terms are proportional to the Wilsonian gauge threshold correctionswhich depend only on the Kahler moduli via instanton corrections (for N = 1sectors they are even vanishing in the setup at hand) This seems to suggest thatthere should better be no UminusθI

I dependence in the matter field Kahler metricsie ξprime = 0

Moreover in analogy to the heterotic string we expect that for the rangeminus1 le θJ le 1 the exponent of TJ runs over the range [minus1 0] This conditionwould fix ζ prime = plusmn12

Let us summarise the conclusions we have drawn from requiring holomorphyof the Wilsonian gauge kinetic function First we provided arguments that theKahler metric for N = 1 chiral matter fields in intersecting D6-brane models isof the following form

Kabij = δij S

minus 14

3prod

J=1

Uminus 1

4J T

minus( 12plusmn 1

2sign(Iab) θ

Jab)

J

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (524)

15

where supersymmetry of course requiressum3

I=1 θIab = 0 Second the holomorphic

gauge kinetic function (on the background considered) only receives correctionsfrom N = 2 open string sectors and the one-loop correction takes on the followingform

f (1)a = minus

sum

b

Nb |IJabIKab |4π2

ln η(i T cI ) I 6= J 6= K 6= I (525)

where the sum only runs over branes b which lie on top of brane a in exactly onetorus denoted by I Therefore the results for the gauge threshold corrections andthe matter field Kahler metrics are consistent both with the non-renormalisationtheorem from section 2 and the Kaplunovsky-Louis formula (51) Clearly itwould be interesting along the lines of [34] to carry out a string amplitude com-putation to fix the free coefficient in the ansatz (511) and see whether our indirectarguments are correct

6 Holomorphic E2-instanton amplitudes

Space-time instantons are given also by D-branes which in this case are EuclideanD2-branes (so-called E2-branes) wrapping three-cycles Ξ in the CalabindashYau sothat they are point-like in four-dimensional Minkowski space Such instantonscan contribute to the holomorphic superpotential and gauge kinetic functions onlyif they preserve half of the N = 1 supersymmetry This means that the instantonmeasure must contain a factor d4x d2θ Let us first clarify an important aspectof this half-BPS condition In the second part of this section we then revisitthe computation of contributions of such instantons to the superpotential andalso clarify some issues concerned with the appearing one-loop determinants Inthe third and fourth part we investigate under which conditions such stringinstantons can also contribute to the gauge kinetic functions and FI-terms

61 Half-BPS instantons

As has been explained in [24 18 20] just wrapping an E2-instanton around arigid sLag three-cycle in the Calabi-Yau gives four bosonic and four fermioniczero modes The vertex operators for the latter are

V(minus12)θ (z) = θα e

minusϕ(z)2 (z)Sα(z) Σh= 3

8q= 3

2(z) (61)

and

V(minus12)

θ(z) = θα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 3

2(z) (62)

Therefore if the instanton is not invariant under the orientifold projection onestill has four instead of the desired two fermionic zero modes Thus only by

16

placing the E2-brane in a position invariant under Ωσ does one have a chanceto get rid of the two additional zero modes θ For so called O(n) instantons onecan see that the zero modes xmicro θ are symmetrised and the mode θ gets anti-symmetrised For the opposite projection ie for USp(2n) instantons the zeromodes xmicro θ are anti-symmetrised and the mode θ gets symmetrised Thereforeone can only get the simple d4x d2θ instanton measure for a single O(1) instanton

62 Superpotential contributions

In order to contribute to the superpotential we also require that there do notarise any further zero modes from E2-E2 open strings so that the three-cycleΞ should be rigid ie b1(Ξ) = 0 Therefore considering an E2-instanton in anintersecting brane configuration additional zero modes can only arise from theintersection of the instanton Ξ with D6-branes Πa There are Na [ΞcapΠa]

+ chiralfermionic zero modes λaI and Na [Ξ cap Πa]

minus anti-chiral ones λaJ 2

For its presentation it is useful to introduce the short-hand notation

Φak bk [~xk] = Φak xk1middot Φxk1xk2

middot Φxk2xk3middot middot Φxknminus1xkn

middot Φxkn(k)bk (63)

for the chain-product of open string vertex operators Here we define Φak bk [~0] =Φak bk

To extract the superpotential one can probe it by evaluating an appropriatematter field correlator in the instanton background The CFT allows one tocompute it in physical normalisation which combines the superpotential part Ywith the matter field Kahler metrics like

〈Φa1b1 middot middot ΦaM bM 〉E2minusinst =e

K

2 YΦa1b1ΦaMbMradic

Ka1b1 middot middotKaM bM

(64)

In [14] a general expression for the single E2-instanton contribution to thecharged matter superpotential was proposed involving the evaluation of the fol-lowing zero mode integral over disc and one-loop open string CFT amplitudes

〈Φa1b1 middot middot ΦaM bM 〉E2 =V3gs

intd4x d2θ

sum

conf

proda

(prod[ΞcapΠa]+

i=1 dλia) (prod[ΞcapΠa]minus

i=1 dλi

a

)

exp(minusSE2) exp (Z prime0(E2)) 〈Φa1b1[~x1]〉λa1 λb1

middot middot 〈ΦaLbL[~xL]〉λaLλbL

(65)

For simplicity we do not consider the case that matter fields are also assignedto string loop diagrams The one-loop contributions are annulus diagrams for

2Here we introduced the physical intersection number between two branes Πa cap Πb whichis the sum of positive [Πa capΠb]

+ and negative [Πa cap Πb]minus intersections

17

open strings with one boundary on the E2-instanton and the other boundary onthe various D6-branes and Mobius diagrams with boundary on the E2-instanton

〈1〉1-loop = Z prime0(E2) =

sumb Z

primeA(E2aD6b) + Z primeM(E2aO6) (66)

Here Z prime means that we only sum over the massive open string states in theloop amplitude as the zero modes are taken care of explicitly It was shownthat these instantonic open string loop diagrams are identical to the one-loopthreshold corrections TA(D6aD6b) Diagrammatically we have the intriguingrelation shown in figure 1 and in figure 2 which holds for the even spin structures3

Fa

Fa x

x

D6aaE2 =D6b D6b

Figure 1 Relation between instantonic one-loop amplitudes and corresponding gaugethreshold corrections

Fa

Fa x

x

D6aaE2 =

O6O6

Figure 2 Relation between instantonic Mobius amplitude and corresponding gaugethreshold corrections

The annulus threshold corrections can be computed leading to

ZA(E2aD6b) =

int infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it)

η3(it)ACY

ab [αβ](it) (67)

and the Mobius strip amplitude for the instanton which as we explained mustbe invariant under the orientifold projection yields

ZM(E2aO6) = plusmnint infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it + 1

2

)

η3(it + 1

2

) ACYaa [α

β](it + 1

2

)(68)

3The contribution of the CP-odd Rminus sector is expected to yield corrections to the θ-angle

18

The overall plus sign is for O(1) instantons reflecting the fact that only for thesethe xmicro and θα zero modes survive the orientifold projection Note that up to theargument the Mobius thresholds are ZA(E2aD6a) Therefore for rigid branesthe massless sector reflects the number of four bosonic and two fermionic zeromodes In section 63 we will discuss the number of zero modes if b1(Ξ) gt 0 Allthese stringy threshold corrections are known to be non-holomorphic Thereforeit is not immediately obvious in which sense the expression (65) is meant andhow one can extract the holomorphic superpotential part Y from it

The CFT disc amplitudes in (65) are also not holomorphic but combinenon-holomorphic Kahler potential contributions and holomorphic superpotentialcontributions in the usual way [38 39 40 41]

〈Φab[~x]〉λaλb=

eK

2 YλaΦax1Φx1x2 ΦxN b λbradicKλaaKax1 KxnbKbλb

(69)

=e

K

2 YλabΦab[x]λbradic

Kλaa Kab[x]Kbλb

(610)

Due to the Kaplunovsky-Louis formula (51) the stringy one-loop amplitudes areknown to include the holomorphic Wilsonian part and contributions from wave-function normalisation Applied to the instanton one-loop amplitudes appearingin Z0(E2a) we write

Z0(E2a) = minus8π2 real(f (1)a )minus ba

2ln

(M2

p

micro2

)minus ca

2Ktree (611)

minus ln

(V3gs

)

tree

+sum

b

|IabNb|2

ln[detKab

]tree

where for the brane and instanton configuration in question the coefficients are

ba =sum

b

|IabNb|2

minus 3 ca =sum

b

|IabNb|2

minus 1 (612)

The constant contributions arise from the Mobius amplitude Inserting (69) and(611) in (65) one realises that the Kahler metrics involving an instanton zeromode and a matter field precisely cancel out so that only the matter metricssurvive as required by the general form (64) Moreover the term exp(K2)comes out just right due to the rule that each disc contains precisely two instantonzero modes The holomorphic piece in (64) can therefore be expressed entirelyin terms of other holomorphic quantities like holomorphic Yukawa couplingsthe holomorphic instanton action and the one-loop holomorphic Wilsonian gaugekinetic function on the E2-brane

YΦa1b1ΦaMbM

=sum

conf

signconf exp(minusSE2)tree exp(minusf (1)

a

)

Yλa1bΦa1b1

[~x1] λb1middot middot Yλa1

bΦaLbL[~xL]λbL

(613)

19

This explicitly shows that knowing the tree-level Kahler potentials computingthe matter field correlator in the instanton background up to one-loop level ings is sufficient to deduce the Wilsonian holomorphic instanton generated super-potential Higher order corrections in gs only come from loop corrections to theKahler potentials

63 Instanton corrections to the gauge kinetic functions

So far we have discussed space-time instanton corrections to the superpotentialThese involved one-loop determinants which are given by annulus vacuum di-agrams with at least one E2 boundary These are related to one-loop gaugethreshold corrections to the gauge theory on a D6-brane wrapping the same cycleas the E2 instanton

Now we can ask what other corrections these space-time instantons can induceBy applying S- and T-dualities to the story of world-sheet instanton correctionsin the heterotic string we expect that there can also be E2-instanton corrections

to the holomorphic gauge kinetic functions In the heterotic case similar to thetopological Type II string such corrections arise from string world-sheets of Eulercharacteristic zero ie here from world-sheets with two boundaries Thereforewe expect such corrections to appear for E2-instantons admitting one complexopen string modulus ie those wrapping a three-cycle with Betti number b1(Ξ) =1

Let us start by discussing the instanton zero mode structure for such a cycleFirst let us provide the form of the vertex operators The bosonic fields in the(minus1) ghost picture are

V (minus1)y (z) = y eminusϕ(z)Σh= 1

2q=plusmn1(z) (614)

which before the orientifold projection are accompanied by the two pairs offermionic zero modes

V (minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 1

2(z) (615)

and

V(minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=+ 1

2(z) (616)

Now one has to distinguish two cases depending on how the anti-holomorphicinvolution σ acts on the open string modulus Y

σ y rarr plusmny (617)

In the case that y is invariant under σ called first kind in the following theorientifold projection acts in the same way as for the 4D fields Xmicro ie thetwo bosonic zero modes y and the two fermionic zero modes micro survive In the

20

other case dubbed second kind the bosonic zero mode is projected out andonly the fermionic modulino zero mode micro survives4 Therefore in the absence ofany additional zero modes for instance from E2-D6 intersections the zero modemeasure in any instanton amplitude assumes the following form

intd4x d2θ d2y d2micro eminusSE2 for σ y rarr y (618)

andintd4x d2θ d2micro eminusSE2 for σ y rarr minusy (619)

As an example consider the set-up in figure 3 with σ yi rarr minusyi Here the

x x x2 31

1 2 3y y y

O6

E2

∆x1 ∆ y3

∆x2

Figure 3 Deformations of an instanton which is invariant under the orientifold pro-jection

deformations ∆x12 are of the first kind and ∆y3 is of the second kindNow it is clear that an instanton with precisely one set of fermionic zero

modes of the second kind and no additional zero modes can generate a correctionto the SU(Na) gauge kinetic function The instanton amplitude takes on thefollowing form

〈Fa(p1)Fa(p2)〉E2 =

intd4x d2θ d2micro exp(minusSE2) exp (Z prime

0(E2)) AF 2a(E2 D6a)

where AF 2a(E2 D6a) is the annulus diagram in figure 4 which absorbs all the

appearing fermionic zero modes and where the gauge boson vertex operators inthe (0)-ghost picture have the usual form

V(0)A (z) = ǫmicro (partmicroX(z) + i(p middot ψ)ψmicro(z)) e

ipmiddotX(z) (620)

4By duality this distinction is related to the two kinds of deformations of genus g curvesstudied in [42] The first kind are the curves moving in families ie transversal deformationof the curve The second kind is related to the deformations coming with the genus g of thecurve

21

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

and therefore only depend on the complex structure moduli At this classicallevel there are no αprime corrections It is an important question about brane stabilitywhether these FI-terms receive perturbative or non-perturbative corrections in gsAgain non-renormalisation theorems say that in the Wilsonian sense one expectsperturbative corrections at most at one-loop

In the following we will be concerned with the terms beyond tree-level appear-ing in (26) (27) and for the FI-terms First we discuss the one-loop thresholdcorrections f 1minusloop

(eminusT c

i

) which also make their appearance in the space-time

instanton generated superpotential W np(eminusUc

I eminusT ci

) Second we will revisit the

computation of stringy one-loop corrections to the FI-terms Finally we willdiscuss fnp

(eminusUc

I eminusT ci

)as well as instanton corrections to the FI-terms

3 One-loop thresholds for intersecting D6-branes

on T6

The purpose of this section is to recall the one-loop results for the gauge thresh-old corrections in intersecting D6-brane models [5 30 6] The gauge couplingconstants of the various gauge group factors Ga in such a model up to one loophave the form

8π2

g2a(micro)=

8π2

g2astring+ba2

ln

(M2

s

micro2

)+

∆a

2 (31)

where ba is the beta function coefficient The first term corresponds to the gaugecoupling constant at the string scale which contains the tree-level gauge couplingas well as the ldquouniversalrdquo contributions at one-loop (see section 52) Thesecontributions are universal in the sense that they originate from a redefinition ofthe dilaton and complex structure moduli at one-loop The redefinition is branestack and therefore gauge group independent However as the gauge couplingsdiffer for the various gauge groups already at tree level this correction effectivelyis gauge group dependent The second term gives the usual one-loop running ofthe coupling constants and the third term denotes the one-loop string thresholdcorrections originating from integrating out massive string excitations The lasttwo terms can be computed as a sum of all annulus and Mobius diagrams withone boundary on brane a in the presence of a background magnetic field in thefour-dimensional space-time

ba ln

(M2

s

micro2

)+∆a =

sum

b

TA(D6aD6b) +sum

bprime

TA(D6aD6bprime) (32)

+ TA(D6aD6aprime) + TM(D6aO6)

Here D6cprime denotes the orientifold image of brane c In an orbifold one also hasto take into account the orbifold images of the branes and orientifold planes

6

The relevant amplitudes for the Z2timesZ2 orbifold have been computed [5 6] Ina sector preserving N = 1 supersymmetry (this means in particular

sumI θ

Iab = 0)

the annulus and Mobius amplitudes are (after subtracting terms which uponsumming over all diagrams vanish due to the tadpole cancellation condition) [6]

TA(D6aD6b) =IabNb

2

[ln

(M2

s

micro2

) 3sum

I=1

sign(θIab)minus

minus ln3prod

I=1

(Γ(|θIab|)

Γ(1minus |θIab|)

)sign(θIab)

minus3sum

I=1

sign(θIab) (ln 2minus γ)

] (33)

TM(D6aO6k) = plusmnIaO6k

[ln

(M2

s

micro2

) 3sum

I=1

sign(θIaO6k)minus

minus ln3prod

I=1

(Γ(2|θIaO6k

|)Γ(1minus 2|θIaO6k

|)

)sign(θIaO6k

)

+3sum

I=1

sign(θIaO6k)(γ minus 3 ln 2)

] (34)

where Iab is the intersection number of branes a and b Nb is the number ofbranes on stack b and πθIab is the intersection angle of branes a and b on the Irsquothtorus Similarly IaO6k denotes the intersection number of brane a and orientifoldplane k and πθIaO6k

their intersection angle The formula for TM is only valid for|θIaO6k

| lt 12 the formulas for other cases look similar [6]In a sector preserving N = 2 supersymmetry one finds [5]

TA(D6aD6b) = Nb|IJab IKab |[ln

(M2

s

micro2

)minus ln |η(i T c

I )|4 minus ln(TI VaI ) + γE minus ln(4π)

]

(35)where I denotes the torus on which the branes lie on top of each other TI itsKahler modulus and T c

I its complexification with TI = real(T cI ) Furthermore

V aI = |nI

a + iuImIa|2uI with uI the complex structure modulus of the torus and

nIam

Ia the wrapping numbers on the Irsquoth torus Note that the moduli dependence

of the one-loop threshold function in the N = 2 sectors is in complete agreementwith the non-renormalisation theorems of section two (see eq (27)) since theholomorphic part of ∆N=2

a is proportional only to ln η(i T cI )

For theN = 1 sectors the one-loop thresholds in a given open string D6-branesector have the following form (specialising to the case θ12ab gt 0 θ3ab lt 0)

∆a = minus ba16π2

ln

[Γ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab)

] (36)

This expression is a non-holomorphic function of the complex structure moduliU cI Hence for the N = 1 sectors the holomorphic one-loop gauge kinetic func-

tion f 1minusloopa

(eminusT c

i

)vanishes The emergence of the non-holomorphic terms in the

one-loop threshold corrections will be further discussed in section 5

7

4 Fayet-Iliopoulos terms

In this section we investigate one-loop corrections to the FI-terms for a U(1)agauge field on the D6a-brane induced by the presence of other branes D6b Suchcorrections for Type I string vacua have already been studied in [31 32] and thecase of intersecting D6-branes has been discussed in [7] Here we are followingessentially the computational technique of [7] The crucial observation is thatthe vertex operator for the auxiliary D-field in the (0)-ghost picture is simply

given by the internal world-sheet U(1) current ie V(0)Da

= JU(1) Therefore the

one-point function of V(0)Da

on the annulus with boundaries a and b can be writtenas

〈VDa〉 = minus i

2πpartν

int infin

0

dt Zab(ν it)|ν=0 (41)

where Zab(ν it) denotes the annulus partition function with insertion of exp(2πiJ0)in the open string sector (ab) where J0 is the zero mode of the U(1) current Inthe case of intersecting D6-branes on a torus preserving N = 1 supersymmetryand after application of the Riemann theta-identities this partition function isgiven by

Zab(ν it) = IabNb(minusi)3π4t2

ϑ1(3ν2 it)

prodI ϑ1(minusν

2+ iθI

2t it)

η3(it)prod

I ϑ1(iθI2t it)

(42)

Using that ϑ1(0 it) = 0 and ϑprime1(ν it)|ν=0 = minus2πη3 one obtains the divergence

〈VDa〉 ≃ IabNb

int infin

0

dt

t2 (43)

which is cancelled by tadpole cancellation in a global model Therefore oncethe D6-branes are supersymmetric at tree-level no FI-term is generated at one-loop level and the system is not destabilised This result is consistent with thecomputations in [31 32]

However this is not the end of the story of the one-loop corrections to FI-terms in intersecting D6-brane models One can also envision that a tree-levelFI-term on a brane D6b induces via a one-loop diagram an FI-term on a braneD6a To proceed we assume that θIb rarr θIb + 2ǫI with

sumǫI = ǫ and compute

〈VDa〉ǫI = minus i

2πpartνpartǫI

int infin

0

dt Zab(ǫI ν it)|ν=0ǫI=0 (44)

with

Zab(ǫI ν it) = IabNb

(minusi)3π4t2

ϑ1(3ν2+ i ǫ

2t it)

prodI ϑ1(minusν

2+ i (ǫ

IminusǫJminusǫK)2

t+ iθI2t it)

η3(it)prod

I ϑ1(iǫIt+ iθI

2t it)

8

The derivative with respect to the supersymmetry breaking parameters ǫI bringsdown one factor of t and it turns out that the result is the same for all partǫI

〈VDa〉ǫ ≃ i IabNb

int infin

0

dt

t

3sum

I=1

ϑprime1ϑ1

(iθI

abt

2 it2

) (45)

This is the same expression as the one-loop threshold corrections TA(D6a D6b)so that at linear order in ǫ we obtain for the FI-terms

(ξa)ǫ = (ǫb minus ǫa) TA(D6a D6b) (46)

Completely analogously one can show that this formula remains true also forN = 2 open string sectors Therefore we would like to propose that such arelation between gauge threshold and one-loop corrections to FI-terms is validfor general intersecting D6-brane models Moreover the Wilsonian part of thethresholds TA(D6a D6b) which we compute in the next section should also bethe Wilsonian part of the correction to the FI-terms

5 Wilsonian gauge kinetic function and σ-model

anomalies

In a supersymmetric gauge theory one can compute the running gauge couplingsga(micro

2) in terms of the gauge kinetic functions fa the Kahler potential K and theKahler metrics of the charged matter fields Kab(micro2) [33 8 9 10 11 12 13]

8π2

g2a(micro2)

= 8π2real(fa) +ba2

ln

(Λ2

micro2

)+ca2K

+T (Ga) ln gminus2a (micro2)minus

sum

r

Ta(r) ln detKr(micro2) (51)

with

ba =sum

r

nrTa(r)minus 3 T (Ga) ca =sum

r

nrTa(r)minus T (Ga) (52)

and Ta(r) = Tr(T 2(a)) (T(a) being the generators of the gauge group Ga) In addi-

tion T (Ga) = Ta(adjGa) and nr is the number of multiplets in the representationr of the gauge group and the sums run over these representations In this contextthe natural cutoff scale for a field theory is the Planck scale ie Λ2 =M2

PlThe left hand side of eq (51) is given by eq (31) which contains the gauge

coupling at the string scale 1g2astring as well as the one-loop string thresholdcorrections ∆a In general ∆a is the sum of a non-holomorphic term plus thereal-part of a holomorphic threshold correction

∆a = ∆nha + real(∆hol

a ) (53)

9

On the right hand side of eq (51) fa denotes the Wilsonian ie holomorphicgauge kinetic function which is given in terms of a holomorphic tree-level functionplus the holomorphic part of the one-loop threshold corrections (cf eq (27))

fa = f treea + f 1minusloop

a

(eminusT c

i

)=sum

I

M IaU

cI +∆hol

a (54)

In addition on the right hand side of eq (51) the non-holomorphic terms propor-tional to the Kahler metric of the moduli K and the matter field Kahler metricsKr are due to the one-loop contributions of massless fields These fields generatenon-local terms in the one-loop effective action which correspond to one-loopnon-invariances under σ-model transformations the so-called σ-model anomalies(Kahler and reparametrisation anomalies)

Matching up all terms in eq (51) essentially means that the σ-model anoma-lies can be cancelled in a two-fold way First by local contributions to the gaugecoupling constant via the one-loop threshold contributions ∆a These terms origi-nate from massive string states The second way to cancel the σ-model anomaliesis due to a field dependent (however gauge group independent) one-loop contribu-tion to the Kahler potential of the chiral moduli fields It implies that some of themoduli fields transform non-trivially under the Kahler transformations and alsounder reparametrisations in moduli space The universal one-loop modificationof the Kahler potential is nothing else than a generalised Green-Schwarz mecha-nism cancelling the σ-model anomalies This is analogous to the Green-Schwarzmechanism which cancels anomalies of physical U(1) gauge fields whereas theσ-model anomalies correspond to unphysical composite gauge connections Ef-fectively it means that the Green-Schwarz mechanism with respect to the σ-modelanomalies can be described by a non-holomorphic one-loop field redefinition ofthe associated tree-level moduli fields

As we will see in type IIA orientifold models these field redefinitions act onthe real parts of the dilaton field S as well as the complex structure moduli UJ

S rarr S + δGS(U T )

UJ rarr UJ + δGSJ (U T ) (55)

These redefined fields are those that determine the gauge coupling constants1g2astring at the string scale Recall that as the tree-level gauge couplings (24)are already gauge group dependent so are these one-loop corrections but theonly dependence arises due to the universal one-loop redefinition of the modulifields It is in this sense that we still call these one-loop corrections to the gaugecouplings universal Note that in heterotic string compactifications the σ-modelGreen-Schwarz mechanism only acts on the heterotic dilaton field

10

51 Holomorphic gauge couplings for toroidal models

In summary equation (51) is to be understood recursively which means that onecan insert the tree-level results into the last three terms of eq (51) In additionone also has to include the universal field redefinition eq (55) in 1g2astring inthe left hand side of eq (51) in order to get a complete matching of all termsin eq (51) as we will demonstrate for the aforementioned toroidal orbifold inthe following For N = 2 sectors the one-loop threshold corrections to the gaugecoupling constant indeed contain a holomorphic Wilsonian term f

(1)a whereas

for N = 1 sectors only the non-holomorphic piece ∆nha is present

Specifically the holomorphic gauge kinetic function can now be determinedby comparing the string theoretical formula (31) for the effective gauge couplingwith the field theoretical one (51) The first thing to notice are the differentcutoff scales appearing in the two formulas One needs to convert one into theother using

M2s

M2Pl

prop exp(2φ4) prop (S U1 U2 U3)minus 1

2 (56)

Here φ4 is the four dimensional dilaton and the complex structure moduli in thesupergravity basis can be expressed in terms of φ4 and the complex structuremoduli uI = RI2RI1 as

S =1

2πeminusφ4

1radicu1 u2 u3

UI =1

2πeminusφ4

radicuJ uKuI

with I 6= J 6= K 6= I (57)

These fields are the real parts of complex scalars of four dimensional chiral mul-tiplets Sc and U c

I As N = 4 super YangndashMills theory is finite one expects the sum of the terms

in (31) proportional to T (Ga) to cancel This is because the only chiral multipletstransforming in the adjoint representation of the gauge group are the open stringmoduli which (on the background considered) assemble themselves into threechiral multiplets thus forming an N = 4 sector together with the gauge fieldsTo show that this cancellation does happen one notices the following Firstlynadjoint = 3 as explained such that there is no term in ba proportional to T (Ga)Secondly

K = minus ln(Sc + Sc)minus

3sum

I=1

ln(U cI + U

c

I)minus3sum

I=1

ln(T cI + T

c

I) (58)

gminus2atree = S

3prod

I=1

nIa minus

3sum

I=1

UI nIam

Jam

Ka I 6= J 6= K 6= I (59)

where TI are the Kahler moduli of the torus and nIa m

Ia are the wrapping numbers

of the brane Finally one needs the matter metric for the open string moduli

11

which can be obtained from the T-dual expression in models with D9- and D5-branes [4] Performing the T-duality which essentially amounts to exchangingKahler and complex structure moduli and converting gauge flux into non-trivialintersection angles for the D6-branes one arrives at (I = 1 3)1

KIij =

δijTIUI

∣∣∣∣∣(nJ

a + iuJ mJa )(n

Ka + iuK m

Ka )

(nIa + iuI mI

a)

∣∣∣∣∣ I 6= J 6= K 6= I (510)

Let us now turn to the fields in the fundamental representation of the gaugegroup Ga in particular to the fields arising from the intersection with one otherstack of branes denoted by b For an N = 1 open string sector the metric forthese fields can be written as [34 4] (see also [35])

Kabij = δij S

minusα

3prod

I=1

Uminus(β+ξ θI

ab)

I Tminus(γ+ζ θI

ab)

I

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (511)

where α β γ ξ and ζ are undetermined constants As θ12ab gt 0 and θ3ab lt 0which is assumed in (511) the intersection number Iab is positive implying that

nf = IabNb (512)

Using Ta(f) =12and relations (52 56 58 511 512) one finds a contribution

to the right hand side of (51) proportional to

IabNb

2

(ln

(M2

s

micro2

)+ (2γ minus 1) ln(T1T2T3) + (2β minus 1

2) ln(U1U2U3) + (2αminus 1

2) lnS

+ζ

3sum

I=1

θIab lnTI + ξ

3sum

I=1

θIab lnUI minus ln

[Γ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab)

]) (513)

Using (36) one finds that the first and the last term exactly reproduce the contri-bution of the last two terms in (31) The terms proportional to ζ and ξ will laterbe shown to constitute the aforementioned universal gauge coupling correctionThe remaining three terms can neither be attributed to such a correction nor canthey be written as the real part of a holomorphic function Thus they cannot bethe one-loop correction to the gauge kinetic function and therefore must vanishThis fixes some of the coefficients in the ansatz (511)

α = β =1

4 γ =

1

2 (514)

1An overall factor involving the wrapping numbers was introduced in this expression in orderto achieve full cancellation This can be done as the expressions used are derived only up tooverall constants [34]

12

The same matching of terms appears between the Mobius diagram plus the annu-lus with boundaries on brane a and its orientifold image and the Kahler metricsfor fields in the symmetric and antisymmetric representation Here one has toreplace θIab and IabNb by θIaaprime = 2θIa and IaaprimeNa in (511) and (33) Apart fromthese replacements the Kahler metric for matter in these representations is alsogiven by (511) with the constants α β γ given in (514)

The corrections to the gauge couplings coming fromN = 2 open string sectorswere seen in the previous section to take on quite a different form They containa term

minus ln |η(i T cI )|4 = minus4Re [ln η(i T c

I )] (515)

which can be written as the real part of a holomorphic function This leads oneto conclude that the gauge kinetic function receives one-loop corrections fromthese sectors Inserting the correct prefactor which from the first term in (35)and the corresponding one in (51) can be seen to be proportional to the betafunction coefficient gives

f (1)a = minusNb |IJab IKab |

4π2ln η(i T c

I ) I 6= J 6= K 6= I (516)

where again I denotes the torus in which the branes lie on top of each other andIJKab are the intersection numbers on the other tori

The term minus ln(TI VaI ) in (35) is not the real part of a holomorphic function

Proceeding as before one finds that the Kahler metric for the hypermultiplet (ortwo chiral multiplets) living at an intersection of branes a and b preserving eightsupercharges must be

KIij =

|nIa + iuI m

Ia|

(UJ UK T J TK)12

I 6= J 6= K 6= I (517)

Apart from the factor in the numerator this is in agreement with the form foundby direct calculations [34 4] The appearance of the numerator is however plau-sible as it also appears in the open string moduli metric and the hypermultipletsunder discussion should feel the Irsquoth torus in the same way

52 Universal threshold corrections

In the following the aforementioned ldquouniversalrdquo gauge coupling corrections willbe discussed They also appear in the heterotic [13] and type I [36 37] string andare related to a redefinition of the dilaton at one-loop [9 8] This stems from thefact that the dilaton really lives in a linear multiplet rather than a chiral one

Our general ansatz for the Kahler metrics for the chiral matter in an N = 1sector contains a factor

3prod

J=1

Uminusξ θJ

ab

J Tminusζ θJ

ab

J (518)

13

which according to (51) appears in the one-loop correction to the gauge couplingconstant Neither is this term reproduced in the string one-loop calculation ofthe coupling nor can it be written as a correction to the holomorphic gaugekinetic function Therefore as is familiar from gauge threshold computationsthere remains the possibility that it can be absorbed into a one-loop correctionto the S and UI chiral superfields In the following we require that such a gaugegroup factor independent universal correction is possible and see how this fixesthe parameters in (518)

The first observation is that in order to get something gauge group indepen-dent the factor (518) actually must have the following form

3prod

J=1

Uminusξprime sign(Iab)θ

Jab

J Tminusζprime sign(Iab)θ

Jab

J (519)

with ξprime and ζ prime independent of the brane For the metrics of fields transformingin the symmetric or antisymmetric representation of the gauge group one hasto replace φab = φa minus φb by φaaprime = 2φa and sign(Iab) by sign(Iaaprime minus IaO6) orsign(Iaaprime + IaO6) respectively Then one computes (K prime denotes the factor (518519) appearing in the full Kahler metric K)

sum

r

Ta(r) ln detKprimer = |Iab|Nb

2ln

[3prod

J=1

Uminusξprime sign(Iab) θ

Jab

J Tminusζprime sign(Iab) θ

Jab

J

](520)

+|Iabprime |Nb

2ln

[3prod

J=1

Uminusξprime sign(Iabprime) θ

Jabprime

J Tminusζprime sign(Iabprime ) θ

Jabprime

J

]

+ Na+22

|IaaprimeminusIaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(IaaprimeminusIaO6) θ

Ja

J Tminus2ζprime sign(IaaprimeminusIaO6) θ

Ja

J

]

+ Naminus22

|Iaaprime+IaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(Iaaprime+IaO6) θ

Ja

J Tminus2ζprime sign(Iaaprime+IaO6) θ

Ja

J

]

After a few steps using |Iab| sign(Iab) = Iab and the tadpole cancellation condi-tion this can be brought to the simple form

sum

r

Ta(r) ln detKprimer = minusn1

an2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

minus3sum

J=1

nJam

Ka m

La

[sum

b

NbmJb n

Kb n

Lb

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

J 6= K 6= L 6= J (521)

Therefore these corrections have precisely the form required for them to be iden-tified with the one-loop correction between the linear superfields appearing in

14

string theory and the chiral superfields used in the supergravity description

SL = S minus 1

8π2

sum

b

Nbm1bm

2bm

3b

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

ULJ = UJ +

1

8π2

sum

b

NbmJb n

Kb n

Lb

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

J 6= K 6= L 6= J (522)

In contrast to eq (59) where the tree-level gauge couplings are determined theone-loop gauge couplings at the string scale have to include the redefined fieldsSL and UL

gminus2astring = SL

3prod

I=1

nIa minus

3sum

I=1

ULI n

Iam

Jam

Ka (523)

In contrast to all (to us) known cases studied in the literature for ξprime 6= 0 thefields which are corrected ie the moduli S and UI also appear in the one-loopredefinition Let us propose an argument why such corrections might be expectedto be absent Due to the anomalous U(1) gauge symmetries the chiral superfieldsS and UI participate in the Green-Schwarz mechanism and therefore transformnon-trivially under U(1) gauge transformations This implies that in order tobe gauge invariant the one-loop corrections in (522) proportional to lnUI mustbe extended in the usual way by UI rarr UI + δaGSVa Computing the resultingFI-terms via the supergravity formula ξa2g

2a = partKpartVa|Va=0 gives besides the

tree-level result depending on SL ULI a one-loop contribution proportional to

ξprimesum

b ξ(0)b φJ

b UJ This has an extra dependence on the complex structure moduliUI However for intersecting D6-branes we have seen that the Wilsonian (super-gravity) FI-Terms are proportional to the Wilsonian gauge threshold correctionswhich depend only on the Kahler moduli via instanton corrections (for N = 1sectors they are even vanishing in the setup at hand) This seems to suggest thatthere should better be no UminusθI

I dependence in the matter field Kahler metricsie ξprime = 0

Moreover in analogy to the heterotic string we expect that for the rangeminus1 le θJ le 1 the exponent of TJ runs over the range [minus1 0] This conditionwould fix ζ prime = plusmn12

Let us summarise the conclusions we have drawn from requiring holomorphyof the Wilsonian gauge kinetic function First we provided arguments that theKahler metric for N = 1 chiral matter fields in intersecting D6-brane models isof the following form

Kabij = δij S

minus 14

3prod

J=1

Uminus 1

4J T

minus( 12plusmn 1

2sign(Iab) θ

Jab)

J

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (524)

15

where supersymmetry of course requiressum3

I=1 θIab = 0 Second the holomorphic

gauge kinetic function (on the background considered) only receives correctionsfrom N = 2 open string sectors and the one-loop correction takes on the followingform

f (1)a = minus

sum

b

Nb |IJabIKab |4π2

ln η(i T cI ) I 6= J 6= K 6= I (525)

where the sum only runs over branes b which lie on top of brane a in exactly onetorus denoted by I Therefore the results for the gauge threshold corrections andthe matter field Kahler metrics are consistent both with the non-renormalisationtheorem from section 2 and the Kaplunovsky-Louis formula (51) Clearly itwould be interesting along the lines of [34] to carry out a string amplitude com-putation to fix the free coefficient in the ansatz (511) and see whether our indirectarguments are correct

6 Holomorphic E2-instanton amplitudes

Space-time instantons are given also by D-branes which in this case are EuclideanD2-branes (so-called E2-branes) wrapping three-cycles Ξ in the CalabindashYau sothat they are point-like in four-dimensional Minkowski space Such instantonscan contribute to the holomorphic superpotential and gauge kinetic functions onlyif they preserve half of the N = 1 supersymmetry This means that the instantonmeasure must contain a factor d4x d2θ Let us first clarify an important aspectof this half-BPS condition In the second part of this section we then revisitthe computation of contributions of such instantons to the superpotential andalso clarify some issues concerned with the appearing one-loop determinants Inthe third and fourth part we investigate under which conditions such stringinstantons can also contribute to the gauge kinetic functions and FI-terms

61 Half-BPS instantons

As has been explained in [24 18 20] just wrapping an E2-instanton around arigid sLag three-cycle in the Calabi-Yau gives four bosonic and four fermioniczero modes The vertex operators for the latter are

V(minus12)θ (z) = θα e

minusϕ(z)2 (z)Sα(z) Σh= 3

8q= 3

2(z) (61)

and

V(minus12)

θ(z) = θα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 3

2(z) (62)

Therefore if the instanton is not invariant under the orientifold projection onestill has four instead of the desired two fermionic zero modes Thus only by

16

placing the E2-brane in a position invariant under Ωσ does one have a chanceto get rid of the two additional zero modes θ For so called O(n) instantons onecan see that the zero modes xmicro θ are symmetrised and the mode θ gets anti-symmetrised For the opposite projection ie for USp(2n) instantons the zeromodes xmicro θ are anti-symmetrised and the mode θ gets symmetrised Thereforeone can only get the simple d4x d2θ instanton measure for a single O(1) instanton

62 Superpotential contributions

In order to contribute to the superpotential we also require that there do notarise any further zero modes from E2-E2 open strings so that the three-cycleΞ should be rigid ie b1(Ξ) = 0 Therefore considering an E2-instanton in anintersecting brane configuration additional zero modes can only arise from theintersection of the instanton Ξ with D6-branes Πa There are Na [ΞcapΠa]

+ chiralfermionic zero modes λaI and Na [Ξ cap Πa]

minus anti-chiral ones λaJ 2

For its presentation it is useful to introduce the short-hand notation

Φak bk [~xk] = Φak xk1middot Φxk1xk2

middot Φxk2xk3middot middot Φxknminus1xkn

middot Φxkn(k)bk (63)

for the chain-product of open string vertex operators Here we define Φak bk [~0] =Φak bk

To extract the superpotential one can probe it by evaluating an appropriatematter field correlator in the instanton background The CFT allows one tocompute it in physical normalisation which combines the superpotential part Ywith the matter field Kahler metrics like

〈Φa1b1 middot middot ΦaM bM 〉E2minusinst =e

K

2 YΦa1b1ΦaMbMradic

Ka1b1 middot middotKaM bM

(64)

In [14] a general expression for the single E2-instanton contribution to thecharged matter superpotential was proposed involving the evaluation of the fol-lowing zero mode integral over disc and one-loop open string CFT amplitudes

〈Φa1b1 middot middot ΦaM bM 〉E2 =V3gs

intd4x d2θ

sum

conf

proda

(prod[ΞcapΠa]+

i=1 dλia) (prod[ΞcapΠa]minus

i=1 dλi

a

)

exp(minusSE2) exp (Z prime0(E2)) 〈Φa1b1[~x1]〉λa1 λb1

middot middot 〈ΦaLbL[~xL]〉λaLλbL

(65)

For simplicity we do not consider the case that matter fields are also assignedto string loop diagrams The one-loop contributions are annulus diagrams for

2Here we introduced the physical intersection number between two branes Πa cap Πb whichis the sum of positive [Πa capΠb]

+ and negative [Πa cap Πb]minus intersections

17

open strings with one boundary on the E2-instanton and the other boundary onthe various D6-branes and Mobius diagrams with boundary on the E2-instanton

〈1〉1-loop = Z prime0(E2) =

sumb Z

primeA(E2aD6b) + Z primeM(E2aO6) (66)

Here Z prime means that we only sum over the massive open string states in theloop amplitude as the zero modes are taken care of explicitly It was shownthat these instantonic open string loop diagrams are identical to the one-loopthreshold corrections TA(D6aD6b) Diagrammatically we have the intriguingrelation shown in figure 1 and in figure 2 which holds for the even spin structures3

Fa

Fa x

x

D6aaE2 =D6b D6b

Figure 1 Relation between instantonic one-loop amplitudes and corresponding gaugethreshold corrections

Fa

Fa x

x

D6aaE2 =

O6O6

Figure 2 Relation between instantonic Mobius amplitude and corresponding gaugethreshold corrections

The annulus threshold corrections can be computed leading to

ZA(E2aD6b) =

int infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it)

η3(it)ACY

ab [αβ](it) (67)

and the Mobius strip amplitude for the instanton which as we explained mustbe invariant under the orientifold projection yields

ZM(E2aO6) = plusmnint infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it + 1

2

)

η3(it + 1

2

) ACYaa [α

β](it + 1

2

)(68)

3The contribution of the CP-odd Rminus sector is expected to yield corrections to the θ-angle

18

The overall plus sign is for O(1) instantons reflecting the fact that only for thesethe xmicro and θα zero modes survive the orientifold projection Note that up to theargument the Mobius thresholds are ZA(E2aD6a) Therefore for rigid branesthe massless sector reflects the number of four bosonic and two fermionic zeromodes In section 63 we will discuss the number of zero modes if b1(Ξ) gt 0 Allthese stringy threshold corrections are known to be non-holomorphic Thereforeit is not immediately obvious in which sense the expression (65) is meant andhow one can extract the holomorphic superpotential part Y from it

The CFT disc amplitudes in (65) are also not holomorphic but combinenon-holomorphic Kahler potential contributions and holomorphic superpotentialcontributions in the usual way [38 39 40 41]

〈Φab[~x]〉λaλb=

eK

2 YλaΦax1Φx1x2 ΦxN b λbradicKλaaKax1 KxnbKbλb

(69)

=e

K

2 YλabΦab[x]λbradic

Kλaa Kab[x]Kbλb

(610)

Due to the Kaplunovsky-Louis formula (51) the stringy one-loop amplitudes areknown to include the holomorphic Wilsonian part and contributions from wave-function normalisation Applied to the instanton one-loop amplitudes appearingin Z0(E2a) we write

Z0(E2a) = minus8π2 real(f (1)a )minus ba

2ln

(M2

p

micro2

)minus ca

2Ktree (611)

minus ln

(V3gs

)

tree

+sum

b

|IabNb|2

ln[detKab

]tree

where for the brane and instanton configuration in question the coefficients are

ba =sum

b

|IabNb|2

minus 3 ca =sum

b

|IabNb|2

minus 1 (612)

The constant contributions arise from the Mobius amplitude Inserting (69) and(611) in (65) one realises that the Kahler metrics involving an instanton zeromode and a matter field precisely cancel out so that only the matter metricssurvive as required by the general form (64) Moreover the term exp(K2)comes out just right due to the rule that each disc contains precisely two instantonzero modes The holomorphic piece in (64) can therefore be expressed entirelyin terms of other holomorphic quantities like holomorphic Yukawa couplingsthe holomorphic instanton action and the one-loop holomorphic Wilsonian gaugekinetic function on the E2-brane

YΦa1b1ΦaMbM

=sum

conf

signconf exp(minusSE2)tree exp(minusf (1)

a

)

Yλa1bΦa1b1

[~x1] λb1middot middot Yλa1

bΦaLbL[~xL]λbL

(613)

19

This explicitly shows that knowing the tree-level Kahler potentials computingthe matter field correlator in the instanton background up to one-loop level ings is sufficient to deduce the Wilsonian holomorphic instanton generated super-potential Higher order corrections in gs only come from loop corrections to theKahler potentials

63 Instanton corrections to the gauge kinetic functions

So far we have discussed space-time instanton corrections to the superpotentialThese involved one-loop determinants which are given by annulus vacuum di-agrams with at least one E2 boundary These are related to one-loop gaugethreshold corrections to the gauge theory on a D6-brane wrapping the same cycleas the E2 instanton

Now we can ask what other corrections these space-time instantons can induceBy applying S- and T-dualities to the story of world-sheet instanton correctionsin the heterotic string we expect that there can also be E2-instanton corrections

to the holomorphic gauge kinetic functions In the heterotic case similar to thetopological Type II string such corrections arise from string world-sheets of Eulercharacteristic zero ie here from world-sheets with two boundaries Thereforewe expect such corrections to appear for E2-instantons admitting one complexopen string modulus ie those wrapping a three-cycle with Betti number b1(Ξ) =1

Let us start by discussing the instanton zero mode structure for such a cycleFirst let us provide the form of the vertex operators The bosonic fields in the(minus1) ghost picture are

V (minus1)y (z) = y eminusϕ(z)Σh= 1

2q=plusmn1(z) (614)

which before the orientifold projection are accompanied by the two pairs offermionic zero modes

V (minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 1

2(z) (615)

and

V(minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=+ 1

2(z) (616)

Now one has to distinguish two cases depending on how the anti-holomorphicinvolution σ acts on the open string modulus Y

σ y rarr plusmny (617)

In the case that y is invariant under σ called first kind in the following theorientifold projection acts in the same way as for the 4D fields Xmicro ie thetwo bosonic zero modes y and the two fermionic zero modes micro survive In the

20

other case dubbed second kind the bosonic zero mode is projected out andonly the fermionic modulino zero mode micro survives4 Therefore in the absence ofany additional zero modes for instance from E2-D6 intersections the zero modemeasure in any instanton amplitude assumes the following form

intd4x d2θ d2y d2micro eminusSE2 for σ y rarr y (618)

andintd4x d2θ d2micro eminusSE2 for σ y rarr minusy (619)

As an example consider the set-up in figure 3 with σ yi rarr minusyi Here the

x x x2 31

1 2 3y y y

O6

E2

∆x1 ∆ y3

∆x2

Figure 3 Deformations of an instanton which is invariant under the orientifold pro-jection

deformations ∆x12 are of the first kind and ∆y3 is of the second kindNow it is clear that an instanton with precisely one set of fermionic zero

modes of the second kind and no additional zero modes can generate a correctionto the SU(Na) gauge kinetic function The instanton amplitude takes on thefollowing form

〈Fa(p1)Fa(p2)〉E2 =

intd4x d2θ d2micro exp(minusSE2) exp (Z prime

0(E2)) AF 2a(E2 D6a)

where AF 2a(E2 D6a) is the annulus diagram in figure 4 which absorbs all the

appearing fermionic zero modes and where the gauge boson vertex operators inthe (0)-ghost picture have the usual form

V(0)A (z) = ǫmicro (partmicroX(z) + i(p middot ψ)ψmicro(z)) e

ipmiddotX(z) (620)

4By duality this distinction is related to the two kinds of deformations of genus g curvesstudied in [42] The first kind are the curves moving in families ie transversal deformationof the curve The second kind is related to the deformations coming with the genus g of thecurve

21

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

The relevant amplitudes for the Z2timesZ2 orbifold have been computed [5 6] Ina sector preserving N = 1 supersymmetry (this means in particular

sumI θ

Iab = 0)

the annulus and Mobius amplitudes are (after subtracting terms which uponsumming over all diagrams vanish due to the tadpole cancellation condition) [6]

TA(D6aD6b) =IabNb

2

[ln

(M2

s

micro2

) 3sum

I=1

sign(θIab)minus

minus ln3prod

I=1

(Γ(|θIab|)

Γ(1minus |θIab|)

)sign(θIab)

minus3sum

I=1

sign(θIab) (ln 2minus γ)

] (33)

TM(D6aO6k) = plusmnIaO6k

[ln

(M2

s

micro2

) 3sum

I=1

sign(θIaO6k)minus

minus ln3prod

I=1

(Γ(2|θIaO6k

|)Γ(1minus 2|θIaO6k

|)

)sign(θIaO6k

)

+3sum

I=1

sign(θIaO6k)(γ minus 3 ln 2)

] (34)

where Iab is the intersection number of branes a and b Nb is the number ofbranes on stack b and πθIab is the intersection angle of branes a and b on the Irsquothtorus Similarly IaO6k denotes the intersection number of brane a and orientifoldplane k and πθIaO6k

their intersection angle The formula for TM is only valid for|θIaO6k

| lt 12 the formulas for other cases look similar [6]In a sector preserving N = 2 supersymmetry one finds [5]

TA(D6aD6b) = Nb|IJab IKab |[ln

(M2

s

micro2

)minus ln |η(i T c

I )|4 minus ln(TI VaI ) + γE minus ln(4π)

]

(35)where I denotes the torus on which the branes lie on top of each other TI itsKahler modulus and T c

I its complexification with TI = real(T cI ) Furthermore

V aI = |nI

a + iuImIa|2uI with uI the complex structure modulus of the torus and

nIam

Ia the wrapping numbers on the Irsquoth torus Note that the moduli dependence

of the one-loop threshold function in the N = 2 sectors is in complete agreementwith the non-renormalisation theorems of section two (see eq (27)) since theholomorphic part of ∆N=2

a is proportional only to ln η(i T cI )

For theN = 1 sectors the one-loop thresholds in a given open string D6-branesector have the following form (specialising to the case θ12ab gt 0 θ3ab lt 0)

∆a = minus ba16π2

ln

[Γ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab)

] (36)

This expression is a non-holomorphic function of the complex structure moduliU cI Hence for the N = 1 sectors the holomorphic one-loop gauge kinetic func-

tion f 1minusloopa

(eminusT c

i

)vanishes The emergence of the non-holomorphic terms in the

one-loop threshold corrections will be further discussed in section 5

7

4 Fayet-Iliopoulos terms

In this section we investigate one-loop corrections to the FI-terms for a U(1)agauge field on the D6a-brane induced by the presence of other branes D6b Suchcorrections for Type I string vacua have already been studied in [31 32] and thecase of intersecting D6-branes has been discussed in [7] Here we are followingessentially the computational technique of [7] The crucial observation is thatthe vertex operator for the auxiliary D-field in the (0)-ghost picture is simply

given by the internal world-sheet U(1) current ie V(0)Da

= JU(1) Therefore the

one-point function of V(0)Da

on the annulus with boundaries a and b can be writtenas

〈VDa〉 = minus i

2πpartν

int infin

0

dt Zab(ν it)|ν=0 (41)

where Zab(ν it) denotes the annulus partition function with insertion of exp(2πiJ0)in the open string sector (ab) where J0 is the zero mode of the U(1) current Inthe case of intersecting D6-branes on a torus preserving N = 1 supersymmetryand after application of the Riemann theta-identities this partition function isgiven by

Zab(ν it) = IabNb(minusi)3π4t2

ϑ1(3ν2 it)

prodI ϑ1(minusν

2+ iθI

2t it)

η3(it)prod

I ϑ1(iθI2t it)

(42)

Using that ϑ1(0 it) = 0 and ϑprime1(ν it)|ν=0 = minus2πη3 one obtains the divergence

〈VDa〉 ≃ IabNb

int infin

0

dt

t2 (43)

which is cancelled by tadpole cancellation in a global model Therefore oncethe D6-branes are supersymmetric at tree-level no FI-term is generated at one-loop level and the system is not destabilised This result is consistent with thecomputations in [31 32]

However this is not the end of the story of the one-loop corrections to FI-terms in intersecting D6-brane models One can also envision that a tree-levelFI-term on a brane D6b induces via a one-loop diagram an FI-term on a braneD6a To proceed we assume that θIb rarr θIb + 2ǫI with

sumǫI = ǫ and compute

〈VDa〉ǫI = minus i

2πpartνpartǫI

int infin

0

dt Zab(ǫI ν it)|ν=0ǫI=0 (44)

with

Zab(ǫI ν it) = IabNb

(minusi)3π4t2

ϑ1(3ν2+ i ǫ

2t it)

prodI ϑ1(minusν

2+ i (ǫ

IminusǫJminusǫK)2

t+ iθI2t it)

η3(it)prod

I ϑ1(iǫIt+ iθI

2t it)

8

The derivative with respect to the supersymmetry breaking parameters ǫI bringsdown one factor of t and it turns out that the result is the same for all partǫI

〈VDa〉ǫ ≃ i IabNb

int infin

0

dt

t

3sum

I=1

ϑprime1ϑ1

(iθI

abt

2 it2

) (45)

This is the same expression as the one-loop threshold corrections TA(D6a D6b)so that at linear order in ǫ we obtain for the FI-terms

(ξa)ǫ = (ǫb minus ǫa) TA(D6a D6b) (46)

Completely analogously one can show that this formula remains true also forN = 2 open string sectors Therefore we would like to propose that such arelation between gauge threshold and one-loop corrections to FI-terms is validfor general intersecting D6-brane models Moreover the Wilsonian part of thethresholds TA(D6a D6b) which we compute in the next section should also bethe Wilsonian part of the correction to the FI-terms

5 Wilsonian gauge kinetic function and σ-model

anomalies

In a supersymmetric gauge theory one can compute the running gauge couplingsga(micro

2) in terms of the gauge kinetic functions fa the Kahler potential K and theKahler metrics of the charged matter fields Kab(micro2) [33 8 9 10 11 12 13]

8π2

g2a(micro2)

= 8π2real(fa) +ba2

ln

(Λ2

micro2

)+ca2K

+T (Ga) ln gminus2a (micro2)minus

sum

r

Ta(r) ln detKr(micro2) (51)

with

ba =sum

r

nrTa(r)minus 3 T (Ga) ca =sum

r

nrTa(r)minus T (Ga) (52)

and Ta(r) = Tr(T 2(a)) (T(a) being the generators of the gauge group Ga) In addi-

tion T (Ga) = Ta(adjGa) and nr is the number of multiplets in the representationr of the gauge group and the sums run over these representations In this contextthe natural cutoff scale for a field theory is the Planck scale ie Λ2 =M2

PlThe left hand side of eq (51) is given by eq (31) which contains the gauge

coupling at the string scale 1g2astring as well as the one-loop string thresholdcorrections ∆a In general ∆a is the sum of a non-holomorphic term plus thereal-part of a holomorphic threshold correction

∆a = ∆nha + real(∆hol

a ) (53)

9

On the right hand side of eq (51) fa denotes the Wilsonian ie holomorphicgauge kinetic function which is given in terms of a holomorphic tree-level functionplus the holomorphic part of the one-loop threshold corrections (cf eq (27))

fa = f treea + f 1minusloop

a

(eminusT c

i

)=sum

I

M IaU

cI +∆hol

a (54)

In addition on the right hand side of eq (51) the non-holomorphic terms propor-tional to the Kahler metric of the moduli K and the matter field Kahler metricsKr are due to the one-loop contributions of massless fields These fields generatenon-local terms in the one-loop effective action which correspond to one-loopnon-invariances under σ-model transformations the so-called σ-model anomalies(Kahler and reparametrisation anomalies)

Matching up all terms in eq (51) essentially means that the σ-model anoma-lies can be cancelled in a two-fold way First by local contributions to the gaugecoupling constant via the one-loop threshold contributions ∆a These terms origi-nate from massive string states The second way to cancel the σ-model anomaliesis due to a field dependent (however gauge group independent) one-loop contribu-tion to the Kahler potential of the chiral moduli fields It implies that some of themoduli fields transform non-trivially under the Kahler transformations and alsounder reparametrisations in moduli space The universal one-loop modificationof the Kahler potential is nothing else than a generalised Green-Schwarz mecha-nism cancelling the σ-model anomalies This is analogous to the Green-Schwarzmechanism which cancels anomalies of physical U(1) gauge fields whereas theσ-model anomalies correspond to unphysical composite gauge connections Ef-fectively it means that the Green-Schwarz mechanism with respect to the σ-modelanomalies can be described by a non-holomorphic one-loop field redefinition ofthe associated tree-level moduli fields

As we will see in type IIA orientifold models these field redefinitions act onthe real parts of the dilaton field S as well as the complex structure moduli UJ

S rarr S + δGS(U T )

UJ rarr UJ + δGSJ (U T ) (55)

These redefined fields are those that determine the gauge coupling constants1g2astring at the string scale Recall that as the tree-level gauge couplings (24)are already gauge group dependent so are these one-loop corrections but theonly dependence arises due to the universal one-loop redefinition of the modulifields It is in this sense that we still call these one-loop corrections to the gaugecouplings universal Note that in heterotic string compactifications the σ-modelGreen-Schwarz mechanism only acts on the heterotic dilaton field

10

51 Holomorphic gauge couplings for toroidal models

In summary equation (51) is to be understood recursively which means that onecan insert the tree-level results into the last three terms of eq (51) In additionone also has to include the universal field redefinition eq (55) in 1g2astring inthe left hand side of eq (51) in order to get a complete matching of all termsin eq (51) as we will demonstrate for the aforementioned toroidal orbifold inthe following For N = 2 sectors the one-loop threshold corrections to the gaugecoupling constant indeed contain a holomorphic Wilsonian term f

(1)a whereas

for N = 1 sectors only the non-holomorphic piece ∆nha is present

Specifically the holomorphic gauge kinetic function can now be determinedby comparing the string theoretical formula (31) for the effective gauge couplingwith the field theoretical one (51) The first thing to notice are the differentcutoff scales appearing in the two formulas One needs to convert one into theother using

M2s

M2Pl

prop exp(2φ4) prop (S U1 U2 U3)minus 1

2 (56)

Here φ4 is the four dimensional dilaton and the complex structure moduli in thesupergravity basis can be expressed in terms of φ4 and the complex structuremoduli uI = RI2RI1 as

S =1

2πeminusφ4

1radicu1 u2 u3

UI =1

2πeminusφ4

radicuJ uKuI

with I 6= J 6= K 6= I (57)

These fields are the real parts of complex scalars of four dimensional chiral mul-tiplets Sc and U c

I As N = 4 super YangndashMills theory is finite one expects the sum of the terms

in (31) proportional to T (Ga) to cancel This is because the only chiral multipletstransforming in the adjoint representation of the gauge group are the open stringmoduli which (on the background considered) assemble themselves into threechiral multiplets thus forming an N = 4 sector together with the gauge fieldsTo show that this cancellation does happen one notices the following Firstlynadjoint = 3 as explained such that there is no term in ba proportional to T (Ga)Secondly

K = minus ln(Sc + Sc)minus

3sum

I=1

ln(U cI + U

c

I)minus3sum

I=1

ln(T cI + T

c

I) (58)

gminus2atree = S

3prod

I=1

nIa minus

3sum

I=1

UI nIam

Jam

Ka I 6= J 6= K 6= I (59)

where TI are the Kahler moduli of the torus and nIa m

Ia are the wrapping numbers

of the brane Finally one needs the matter metric for the open string moduli

11

which can be obtained from the T-dual expression in models with D9- and D5-branes [4] Performing the T-duality which essentially amounts to exchangingKahler and complex structure moduli and converting gauge flux into non-trivialintersection angles for the D6-branes one arrives at (I = 1 3)1

KIij =

δijTIUI

∣∣∣∣∣(nJ

a + iuJ mJa )(n

Ka + iuK m

Ka )

(nIa + iuI mI

a)

∣∣∣∣∣ I 6= J 6= K 6= I (510)

Let us now turn to the fields in the fundamental representation of the gaugegroup Ga in particular to the fields arising from the intersection with one otherstack of branes denoted by b For an N = 1 open string sector the metric forthese fields can be written as [34 4] (see also [35])

Kabij = δij S

minusα

3prod

I=1

Uminus(β+ξ θI

ab)

I Tminus(γ+ζ θI

ab)

I

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (511)

where α β γ ξ and ζ are undetermined constants As θ12ab gt 0 and θ3ab lt 0which is assumed in (511) the intersection number Iab is positive implying that

nf = IabNb (512)

Using Ta(f) =12and relations (52 56 58 511 512) one finds a contribution

to the right hand side of (51) proportional to

IabNb

2

(ln

(M2

s

micro2

)+ (2γ minus 1) ln(T1T2T3) + (2β minus 1

2) ln(U1U2U3) + (2αminus 1

2) lnS

+ζ

3sum

I=1

θIab lnTI + ξ

3sum

I=1

θIab lnUI minus ln

[Γ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab)

]) (513)

Using (36) one finds that the first and the last term exactly reproduce the contri-bution of the last two terms in (31) The terms proportional to ζ and ξ will laterbe shown to constitute the aforementioned universal gauge coupling correctionThe remaining three terms can neither be attributed to such a correction nor canthey be written as the real part of a holomorphic function Thus they cannot bethe one-loop correction to the gauge kinetic function and therefore must vanishThis fixes some of the coefficients in the ansatz (511)

α = β =1

4 γ =

1

2 (514)

1An overall factor involving the wrapping numbers was introduced in this expression in orderto achieve full cancellation This can be done as the expressions used are derived only up tooverall constants [34]

12

The same matching of terms appears between the Mobius diagram plus the annu-lus with boundaries on brane a and its orientifold image and the Kahler metricsfor fields in the symmetric and antisymmetric representation Here one has toreplace θIab and IabNb by θIaaprime = 2θIa and IaaprimeNa in (511) and (33) Apart fromthese replacements the Kahler metric for matter in these representations is alsogiven by (511) with the constants α β γ given in (514)

The corrections to the gauge couplings coming fromN = 2 open string sectorswere seen in the previous section to take on quite a different form They containa term

minus ln |η(i T cI )|4 = minus4Re [ln η(i T c

I )] (515)

which can be written as the real part of a holomorphic function This leads oneto conclude that the gauge kinetic function receives one-loop corrections fromthese sectors Inserting the correct prefactor which from the first term in (35)and the corresponding one in (51) can be seen to be proportional to the betafunction coefficient gives

f (1)a = minusNb |IJab IKab |

4π2ln η(i T c

I ) I 6= J 6= K 6= I (516)

where again I denotes the torus in which the branes lie on top of each other andIJKab are the intersection numbers on the other tori

The term minus ln(TI VaI ) in (35) is not the real part of a holomorphic function

Proceeding as before one finds that the Kahler metric for the hypermultiplet (ortwo chiral multiplets) living at an intersection of branes a and b preserving eightsupercharges must be

KIij =

|nIa + iuI m

Ia|

(UJ UK T J TK)12

I 6= J 6= K 6= I (517)

Apart from the factor in the numerator this is in agreement with the form foundby direct calculations [34 4] The appearance of the numerator is however plau-sible as it also appears in the open string moduli metric and the hypermultipletsunder discussion should feel the Irsquoth torus in the same way

52 Universal threshold corrections

In the following the aforementioned ldquouniversalrdquo gauge coupling corrections willbe discussed They also appear in the heterotic [13] and type I [36 37] string andare related to a redefinition of the dilaton at one-loop [9 8] This stems from thefact that the dilaton really lives in a linear multiplet rather than a chiral one

Our general ansatz for the Kahler metrics for the chiral matter in an N = 1sector contains a factor

3prod

J=1

Uminusξ θJ

ab

J Tminusζ θJ

ab

J (518)

13

which according to (51) appears in the one-loop correction to the gauge couplingconstant Neither is this term reproduced in the string one-loop calculation ofthe coupling nor can it be written as a correction to the holomorphic gaugekinetic function Therefore as is familiar from gauge threshold computationsthere remains the possibility that it can be absorbed into a one-loop correctionto the S and UI chiral superfields In the following we require that such a gaugegroup factor independent universal correction is possible and see how this fixesthe parameters in (518)

The first observation is that in order to get something gauge group indepen-dent the factor (518) actually must have the following form

3prod

J=1

Uminusξprime sign(Iab)θ

Jab

J Tminusζprime sign(Iab)θ

Jab

J (519)

with ξprime and ζ prime independent of the brane For the metrics of fields transformingin the symmetric or antisymmetric representation of the gauge group one hasto replace φab = φa minus φb by φaaprime = 2φa and sign(Iab) by sign(Iaaprime minus IaO6) orsign(Iaaprime + IaO6) respectively Then one computes (K prime denotes the factor (518519) appearing in the full Kahler metric K)

sum

r

Ta(r) ln detKprimer = |Iab|Nb

2ln

[3prod

J=1

Uminusξprime sign(Iab) θ

Jab

J Tminusζprime sign(Iab) θ

Jab

J

](520)

+|Iabprime |Nb

2ln

[3prod

J=1

Uminusξprime sign(Iabprime) θ

Jabprime

J Tminusζprime sign(Iabprime ) θ

Jabprime

J

]

+ Na+22

|IaaprimeminusIaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(IaaprimeminusIaO6) θ

Ja

J Tminus2ζprime sign(IaaprimeminusIaO6) θ

Ja

J

]

+ Naminus22

|Iaaprime+IaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(Iaaprime+IaO6) θ

Ja

J Tminus2ζprime sign(Iaaprime+IaO6) θ

Ja

J

]

After a few steps using |Iab| sign(Iab) = Iab and the tadpole cancellation condi-tion this can be brought to the simple form

sum

r

Ta(r) ln detKprimer = minusn1

an2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

minus3sum

J=1

nJam

Ka m

La

[sum

b

NbmJb n

Kb n

Lb

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

J 6= K 6= L 6= J (521)

Therefore these corrections have precisely the form required for them to be iden-tified with the one-loop correction between the linear superfields appearing in

14

string theory and the chiral superfields used in the supergravity description

SL = S minus 1

8π2

sum

b

Nbm1bm

2bm

3b

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

ULJ = UJ +

1

8π2

sum

b

NbmJb n

Kb n

Lb

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

J 6= K 6= L 6= J (522)

In contrast to eq (59) where the tree-level gauge couplings are determined theone-loop gauge couplings at the string scale have to include the redefined fieldsSL and UL

gminus2astring = SL

3prod

I=1

nIa minus

3sum

I=1

ULI n

Iam

Jam

Ka (523)

In contrast to all (to us) known cases studied in the literature for ξprime 6= 0 thefields which are corrected ie the moduli S and UI also appear in the one-loopredefinition Let us propose an argument why such corrections might be expectedto be absent Due to the anomalous U(1) gauge symmetries the chiral superfieldsS and UI participate in the Green-Schwarz mechanism and therefore transformnon-trivially under U(1) gauge transformations This implies that in order tobe gauge invariant the one-loop corrections in (522) proportional to lnUI mustbe extended in the usual way by UI rarr UI + δaGSVa Computing the resultingFI-terms via the supergravity formula ξa2g

2a = partKpartVa|Va=0 gives besides the

tree-level result depending on SL ULI a one-loop contribution proportional to

ξprimesum

b ξ(0)b φJ

b UJ This has an extra dependence on the complex structure moduliUI However for intersecting D6-branes we have seen that the Wilsonian (super-gravity) FI-Terms are proportional to the Wilsonian gauge threshold correctionswhich depend only on the Kahler moduli via instanton corrections (for N = 1sectors they are even vanishing in the setup at hand) This seems to suggest thatthere should better be no UminusθI

I dependence in the matter field Kahler metricsie ξprime = 0

Moreover in analogy to the heterotic string we expect that for the rangeminus1 le θJ le 1 the exponent of TJ runs over the range [minus1 0] This conditionwould fix ζ prime = plusmn12

Let us summarise the conclusions we have drawn from requiring holomorphyof the Wilsonian gauge kinetic function First we provided arguments that theKahler metric for N = 1 chiral matter fields in intersecting D6-brane models isof the following form

Kabij = δij S

minus 14

3prod

J=1

Uminus 1

4J T

minus( 12plusmn 1

2sign(Iab) θ

Jab)

J

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (524)

15

where supersymmetry of course requiressum3

I=1 θIab = 0 Second the holomorphic

gauge kinetic function (on the background considered) only receives correctionsfrom N = 2 open string sectors and the one-loop correction takes on the followingform

f (1)a = minus

sum

b

Nb |IJabIKab |4π2

ln η(i T cI ) I 6= J 6= K 6= I (525)

where the sum only runs over branes b which lie on top of brane a in exactly onetorus denoted by I Therefore the results for the gauge threshold corrections andthe matter field Kahler metrics are consistent both with the non-renormalisationtheorem from section 2 and the Kaplunovsky-Louis formula (51) Clearly itwould be interesting along the lines of [34] to carry out a string amplitude com-putation to fix the free coefficient in the ansatz (511) and see whether our indirectarguments are correct

6 Holomorphic E2-instanton amplitudes

Space-time instantons are given also by D-branes which in this case are EuclideanD2-branes (so-called E2-branes) wrapping three-cycles Ξ in the CalabindashYau sothat they are point-like in four-dimensional Minkowski space Such instantonscan contribute to the holomorphic superpotential and gauge kinetic functions onlyif they preserve half of the N = 1 supersymmetry This means that the instantonmeasure must contain a factor d4x d2θ Let us first clarify an important aspectof this half-BPS condition In the second part of this section we then revisitthe computation of contributions of such instantons to the superpotential andalso clarify some issues concerned with the appearing one-loop determinants Inthe third and fourth part we investigate under which conditions such stringinstantons can also contribute to the gauge kinetic functions and FI-terms

61 Half-BPS instantons

As has been explained in [24 18 20] just wrapping an E2-instanton around arigid sLag three-cycle in the Calabi-Yau gives four bosonic and four fermioniczero modes The vertex operators for the latter are

V(minus12)θ (z) = θα e

minusϕ(z)2 (z)Sα(z) Σh= 3

8q= 3

2(z) (61)

and

V(minus12)

θ(z) = θα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 3

2(z) (62)

Therefore if the instanton is not invariant under the orientifold projection onestill has four instead of the desired two fermionic zero modes Thus only by

16

placing the E2-brane in a position invariant under Ωσ does one have a chanceto get rid of the two additional zero modes θ For so called O(n) instantons onecan see that the zero modes xmicro θ are symmetrised and the mode θ gets anti-symmetrised For the opposite projection ie for USp(2n) instantons the zeromodes xmicro θ are anti-symmetrised and the mode θ gets symmetrised Thereforeone can only get the simple d4x d2θ instanton measure for a single O(1) instanton

62 Superpotential contributions

In order to contribute to the superpotential we also require that there do notarise any further zero modes from E2-E2 open strings so that the three-cycleΞ should be rigid ie b1(Ξ) = 0 Therefore considering an E2-instanton in anintersecting brane configuration additional zero modes can only arise from theintersection of the instanton Ξ with D6-branes Πa There are Na [ΞcapΠa]

+ chiralfermionic zero modes λaI and Na [Ξ cap Πa]

minus anti-chiral ones λaJ 2

For its presentation it is useful to introduce the short-hand notation

Φak bk [~xk] = Φak xk1middot Φxk1xk2

middot Φxk2xk3middot middot Φxknminus1xkn

middot Φxkn(k)bk (63)

for the chain-product of open string vertex operators Here we define Φak bk [~0] =Φak bk

To extract the superpotential one can probe it by evaluating an appropriatematter field correlator in the instanton background The CFT allows one tocompute it in physical normalisation which combines the superpotential part Ywith the matter field Kahler metrics like

〈Φa1b1 middot middot ΦaM bM 〉E2minusinst =e

K

2 YΦa1b1ΦaMbMradic

Ka1b1 middot middotKaM bM

(64)

In [14] a general expression for the single E2-instanton contribution to thecharged matter superpotential was proposed involving the evaluation of the fol-lowing zero mode integral over disc and one-loop open string CFT amplitudes

〈Φa1b1 middot middot ΦaM bM 〉E2 =V3gs

intd4x d2θ

sum

conf

proda

(prod[ΞcapΠa]+

i=1 dλia) (prod[ΞcapΠa]minus

i=1 dλi

a

)

exp(minusSE2) exp (Z prime0(E2)) 〈Φa1b1[~x1]〉λa1 λb1

middot middot 〈ΦaLbL[~xL]〉λaLλbL

(65)

For simplicity we do not consider the case that matter fields are also assignedto string loop diagrams The one-loop contributions are annulus diagrams for

2Here we introduced the physical intersection number between two branes Πa cap Πb whichis the sum of positive [Πa capΠb]

+ and negative [Πa cap Πb]minus intersections

17

open strings with one boundary on the E2-instanton and the other boundary onthe various D6-branes and Mobius diagrams with boundary on the E2-instanton

〈1〉1-loop = Z prime0(E2) =

sumb Z

primeA(E2aD6b) + Z primeM(E2aO6) (66)

Here Z prime means that we only sum over the massive open string states in theloop amplitude as the zero modes are taken care of explicitly It was shownthat these instantonic open string loop diagrams are identical to the one-loopthreshold corrections TA(D6aD6b) Diagrammatically we have the intriguingrelation shown in figure 1 and in figure 2 which holds for the even spin structures3

Fa

Fa x

x

D6aaE2 =D6b D6b

Figure 1 Relation between instantonic one-loop amplitudes and corresponding gaugethreshold corrections

Fa

Fa x

x

D6aaE2 =

O6O6

Figure 2 Relation between instantonic Mobius amplitude and corresponding gaugethreshold corrections

The annulus threshold corrections can be computed leading to

ZA(E2aD6b) =

int infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it)

η3(it)ACY

ab [αβ](it) (67)

and the Mobius strip amplitude for the instanton which as we explained mustbe invariant under the orientifold projection yields

ZM(E2aO6) = plusmnint infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it + 1

2

)

η3(it + 1

2

) ACYaa [α

β](it + 1

2

)(68)

3The contribution of the CP-odd Rminus sector is expected to yield corrections to the θ-angle

18

The overall plus sign is for O(1) instantons reflecting the fact that only for thesethe xmicro and θα zero modes survive the orientifold projection Note that up to theargument the Mobius thresholds are ZA(E2aD6a) Therefore for rigid branesthe massless sector reflects the number of four bosonic and two fermionic zeromodes In section 63 we will discuss the number of zero modes if b1(Ξ) gt 0 Allthese stringy threshold corrections are known to be non-holomorphic Thereforeit is not immediately obvious in which sense the expression (65) is meant andhow one can extract the holomorphic superpotential part Y from it

The CFT disc amplitudes in (65) are also not holomorphic but combinenon-holomorphic Kahler potential contributions and holomorphic superpotentialcontributions in the usual way [38 39 40 41]

〈Φab[~x]〉λaλb=

eK

2 YλaΦax1Φx1x2 ΦxN b λbradicKλaaKax1 KxnbKbλb

(69)

=e

K

2 YλabΦab[x]λbradic

Kλaa Kab[x]Kbλb

(610)

Due to the Kaplunovsky-Louis formula (51) the stringy one-loop amplitudes areknown to include the holomorphic Wilsonian part and contributions from wave-function normalisation Applied to the instanton one-loop amplitudes appearingin Z0(E2a) we write

Z0(E2a) = minus8π2 real(f (1)a )minus ba

2ln

(M2

p

micro2

)minus ca

2Ktree (611)

minus ln

(V3gs

)

tree

+sum

b

|IabNb|2

ln[detKab

]tree

where for the brane and instanton configuration in question the coefficients are

ba =sum

b

|IabNb|2

minus 3 ca =sum

b

|IabNb|2

minus 1 (612)

The constant contributions arise from the Mobius amplitude Inserting (69) and(611) in (65) one realises that the Kahler metrics involving an instanton zeromode and a matter field precisely cancel out so that only the matter metricssurvive as required by the general form (64) Moreover the term exp(K2)comes out just right due to the rule that each disc contains precisely two instantonzero modes The holomorphic piece in (64) can therefore be expressed entirelyin terms of other holomorphic quantities like holomorphic Yukawa couplingsthe holomorphic instanton action and the one-loop holomorphic Wilsonian gaugekinetic function on the E2-brane

YΦa1b1ΦaMbM

=sum

conf

signconf exp(minusSE2)tree exp(minusf (1)

a

)

Yλa1bΦa1b1

[~x1] λb1middot middot Yλa1

bΦaLbL[~xL]λbL

(613)

19

This explicitly shows that knowing the tree-level Kahler potentials computingthe matter field correlator in the instanton background up to one-loop level ings is sufficient to deduce the Wilsonian holomorphic instanton generated super-potential Higher order corrections in gs only come from loop corrections to theKahler potentials

63 Instanton corrections to the gauge kinetic functions

So far we have discussed space-time instanton corrections to the superpotentialThese involved one-loop determinants which are given by annulus vacuum di-agrams with at least one E2 boundary These are related to one-loop gaugethreshold corrections to the gauge theory on a D6-brane wrapping the same cycleas the E2 instanton

Now we can ask what other corrections these space-time instantons can induceBy applying S- and T-dualities to the story of world-sheet instanton correctionsin the heterotic string we expect that there can also be E2-instanton corrections

to the holomorphic gauge kinetic functions In the heterotic case similar to thetopological Type II string such corrections arise from string world-sheets of Eulercharacteristic zero ie here from world-sheets with two boundaries Thereforewe expect such corrections to appear for E2-instantons admitting one complexopen string modulus ie those wrapping a three-cycle with Betti number b1(Ξ) =1

Let us start by discussing the instanton zero mode structure for such a cycleFirst let us provide the form of the vertex operators The bosonic fields in the(minus1) ghost picture are

V (minus1)y (z) = y eminusϕ(z)Σh= 1

2q=plusmn1(z) (614)

which before the orientifold projection are accompanied by the two pairs offermionic zero modes

V (minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 1

2(z) (615)

and

V(minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=+ 1

2(z) (616)

Now one has to distinguish two cases depending on how the anti-holomorphicinvolution σ acts on the open string modulus Y

σ y rarr plusmny (617)

In the case that y is invariant under σ called first kind in the following theorientifold projection acts in the same way as for the 4D fields Xmicro ie thetwo bosonic zero modes y and the two fermionic zero modes micro survive In the

20

other case dubbed second kind the bosonic zero mode is projected out andonly the fermionic modulino zero mode micro survives4 Therefore in the absence ofany additional zero modes for instance from E2-D6 intersections the zero modemeasure in any instanton amplitude assumes the following form

intd4x d2θ d2y d2micro eminusSE2 for σ y rarr y (618)

andintd4x d2θ d2micro eminusSE2 for σ y rarr minusy (619)

As an example consider the set-up in figure 3 with σ yi rarr minusyi Here the

x x x2 31

1 2 3y y y

O6

E2

∆x1 ∆ y3

∆x2

Figure 3 Deformations of an instanton which is invariant under the orientifold pro-jection

deformations ∆x12 are of the first kind and ∆y3 is of the second kindNow it is clear that an instanton with precisely one set of fermionic zero

modes of the second kind and no additional zero modes can generate a correctionto the SU(Na) gauge kinetic function The instanton amplitude takes on thefollowing form

〈Fa(p1)Fa(p2)〉E2 =

intd4x d2θ d2micro exp(minusSE2) exp (Z prime

0(E2)) AF 2a(E2 D6a)

where AF 2a(E2 D6a) is the annulus diagram in figure 4 which absorbs all the

appearing fermionic zero modes and where the gauge boson vertex operators inthe (0)-ghost picture have the usual form

V(0)A (z) = ǫmicro (partmicroX(z) + i(p middot ψ)ψmicro(z)) e

ipmiddotX(z) (620)

4By duality this distinction is related to the two kinds of deformations of genus g curvesstudied in [42] The first kind are the curves moving in families ie transversal deformationof the curve The second kind is related to the deformations coming with the genus g of thecurve

21

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

4 Fayet-Iliopoulos terms

In this section we investigate one-loop corrections to the FI-terms for a U(1)agauge field on the D6a-brane induced by the presence of other branes D6b Suchcorrections for Type I string vacua have already been studied in [31 32] and thecase of intersecting D6-branes has been discussed in [7] Here we are followingessentially the computational technique of [7] The crucial observation is thatthe vertex operator for the auxiliary D-field in the (0)-ghost picture is simply

given by the internal world-sheet U(1) current ie V(0)Da

= JU(1) Therefore the

one-point function of V(0)Da

on the annulus with boundaries a and b can be writtenas

〈VDa〉 = minus i

2πpartν

int infin

0

dt Zab(ν it)|ν=0 (41)

where Zab(ν it) denotes the annulus partition function with insertion of exp(2πiJ0)in the open string sector (ab) where J0 is the zero mode of the U(1) current Inthe case of intersecting D6-branes on a torus preserving N = 1 supersymmetryand after application of the Riemann theta-identities this partition function isgiven by

Zab(ν it) = IabNb(minusi)3π4t2

ϑ1(3ν2 it)

prodI ϑ1(minusν

2+ iθI

2t it)

η3(it)prod

I ϑ1(iθI2t it)

(42)

Using that ϑ1(0 it) = 0 and ϑprime1(ν it)|ν=0 = minus2πη3 one obtains the divergence

〈VDa〉 ≃ IabNb

int infin

0

dt

t2 (43)

which is cancelled by tadpole cancellation in a global model Therefore oncethe D6-branes are supersymmetric at tree-level no FI-term is generated at one-loop level and the system is not destabilised This result is consistent with thecomputations in [31 32]

However this is not the end of the story of the one-loop corrections to FI-terms in intersecting D6-brane models One can also envision that a tree-levelFI-term on a brane D6b induces via a one-loop diagram an FI-term on a braneD6a To proceed we assume that θIb rarr θIb + 2ǫI with

sumǫI = ǫ and compute

〈VDa〉ǫI = minus i

2πpartνpartǫI

int infin

0

dt Zab(ǫI ν it)|ν=0ǫI=0 (44)

with

Zab(ǫI ν it) = IabNb

(minusi)3π4t2

ϑ1(3ν2+ i ǫ

2t it)

prodI ϑ1(minusν

2+ i (ǫ

IminusǫJminusǫK)2

t+ iθI2t it)

η3(it)prod

I ϑ1(iǫIt+ iθI

2t it)

8

The derivative with respect to the supersymmetry breaking parameters ǫI bringsdown one factor of t and it turns out that the result is the same for all partǫI

〈VDa〉ǫ ≃ i IabNb

int infin

0

dt

t

3sum

I=1

ϑprime1ϑ1

(iθI

abt

2 it2

) (45)

This is the same expression as the one-loop threshold corrections TA(D6a D6b)so that at linear order in ǫ we obtain for the FI-terms

(ξa)ǫ = (ǫb minus ǫa) TA(D6a D6b) (46)

Completely analogously one can show that this formula remains true also forN = 2 open string sectors Therefore we would like to propose that such arelation between gauge threshold and one-loop corrections to FI-terms is validfor general intersecting D6-brane models Moreover the Wilsonian part of thethresholds TA(D6a D6b) which we compute in the next section should also bethe Wilsonian part of the correction to the FI-terms

5 Wilsonian gauge kinetic function and σ-model

anomalies

In a supersymmetric gauge theory one can compute the running gauge couplingsga(micro

2) in terms of the gauge kinetic functions fa the Kahler potential K and theKahler metrics of the charged matter fields Kab(micro2) [33 8 9 10 11 12 13]

8π2

g2a(micro2)

= 8π2real(fa) +ba2

ln

(Λ2

micro2

)+ca2K

+T (Ga) ln gminus2a (micro2)minus

sum

r

Ta(r) ln detKr(micro2) (51)

with

ba =sum

r

nrTa(r)minus 3 T (Ga) ca =sum

r

nrTa(r)minus T (Ga) (52)

and Ta(r) = Tr(T 2(a)) (T(a) being the generators of the gauge group Ga) In addi-

tion T (Ga) = Ta(adjGa) and nr is the number of multiplets in the representationr of the gauge group and the sums run over these representations In this contextthe natural cutoff scale for a field theory is the Planck scale ie Λ2 =M2

PlThe left hand side of eq (51) is given by eq (31) which contains the gauge

coupling at the string scale 1g2astring as well as the one-loop string thresholdcorrections ∆a In general ∆a is the sum of a non-holomorphic term plus thereal-part of a holomorphic threshold correction

∆a = ∆nha + real(∆hol

a ) (53)

9

On the right hand side of eq (51) fa denotes the Wilsonian ie holomorphicgauge kinetic function which is given in terms of a holomorphic tree-level functionplus the holomorphic part of the one-loop threshold corrections (cf eq (27))

fa = f treea + f 1minusloop

a

(eminusT c

i

)=sum

I

M IaU

cI +∆hol

a (54)

In addition on the right hand side of eq (51) the non-holomorphic terms propor-tional to the Kahler metric of the moduli K and the matter field Kahler metricsKr are due to the one-loop contributions of massless fields These fields generatenon-local terms in the one-loop effective action which correspond to one-loopnon-invariances under σ-model transformations the so-called σ-model anomalies(Kahler and reparametrisation anomalies)

Matching up all terms in eq (51) essentially means that the σ-model anoma-lies can be cancelled in a two-fold way First by local contributions to the gaugecoupling constant via the one-loop threshold contributions ∆a These terms origi-nate from massive string states The second way to cancel the σ-model anomaliesis due to a field dependent (however gauge group independent) one-loop contribu-tion to the Kahler potential of the chiral moduli fields It implies that some of themoduli fields transform non-trivially under the Kahler transformations and alsounder reparametrisations in moduli space The universal one-loop modificationof the Kahler potential is nothing else than a generalised Green-Schwarz mecha-nism cancelling the σ-model anomalies This is analogous to the Green-Schwarzmechanism which cancels anomalies of physical U(1) gauge fields whereas theσ-model anomalies correspond to unphysical composite gauge connections Ef-fectively it means that the Green-Schwarz mechanism with respect to the σ-modelanomalies can be described by a non-holomorphic one-loop field redefinition ofthe associated tree-level moduli fields

As we will see in type IIA orientifold models these field redefinitions act onthe real parts of the dilaton field S as well as the complex structure moduli UJ

S rarr S + δGS(U T )

UJ rarr UJ + δGSJ (U T ) (55)

These redefined fields are those that determine the gauge coupling constants1g2astring at the string scale Recall that as the tree-level gauge couplings (24)are already gauge group dependent so are these one-loop corrections but theonly dependence arises due to the universal one-loop redefinition of the modulifields It is in this sense that we still call these one-loop corrections to the gaugecouplings universal Note that in heterotic string compactifications the σ-modelGreen-Schwarz mechanism only acts on the heterotic dilaton field

10

51 Holomorphic gauge couplings for toroidal models

In summary equation (51) is to be understood recursively which means that onecan insert the tree-level results into the last three terms of eq (51) In additionone also has to include the universal field redefinition eq (55) in 1g2astring inthe left hand side of eq (51) in order to get a complete matching of all termsin eq (51) as we will demonstrate for the aforementioned toroidal orbifold inthe following For N = 2 sectors the one-loop threshold corrections to the gaugecoupling constant indeed contain a holomorphic Wilsonian term f

(1)a whereas

for N = 1 sectors only the non-holomorphic piece ∆nha is present

Specifically the holomorphic gauge kinetic function can now be determinedby comparing the string theoretical formula (31) for the effective gauge couplingwith the field theoretical one (51) The first thing to notice are the differentcutoff scales appearing in the two formulas One needs to convert one into theother using

M2s

M2Pl

prop exp(2φ4) prop (S U1 U2 U3)minus 1

2 (56)

Here φ4 is the four dimensional dilaton and the complex structure moduli in thesupergravity basis can be expressed in terms of φ4 and the complex structuremoduli uI = RI2RI1 as

S =1

2πeminusφ4

1radicu1 u2 u3

UI =1

2πeminusφ4

radicuJ uKuI

with I 6= J 6= K 6= I (57)

These fields are the real parts of complex scalars of four dimensional chiral mul-tiplets Sc and U c

I As N = 4 super YangndashMills theory is finite one expects the sum of the terms

in (31) proportional to T (Ga) to cancel This is because the only chiral multipletstransforming in the adjoint representation of the gauge group are the open stringmoduli which (on the background considered) assemble themselves into threechiral multiplets thus forming an N = 4 sector together with the gauge fieldsTo show that this cancellation does happen one notices the following Firstlynadjoint = 3 as explained such that there is no term in ba proportional to T (Ga)Secondly

K = minus ln(Sc + Sc)minus

3sum

I=1

ln(U cI + U

c

I)minus3sum

I=1

ln(T cI + T

c

I) (58)

gminus2atree = S

3prod

I=1

nIa minus

3sum

I=1

UI nIam

Jam

Ka I 6= J 6= K 6= I (59)

where TI are the Kahler moduli of the torus and nIa m

Ia are the wrapping numbers

of the brane Finally one needs the matter metric for the open string moduli

11

which can be obtained from the T-dual expression in models with D9- and D5-branes [4] Performing the T-duality which essentially amounts to exchangingKahler and complex structure moduli and converting gauge flux into non-trivialintersection angles for the D6-branes one arrives at (I = 1 3)1

KIij =

δijTIUI

∣∣∣∣∣(nJ

a + iuJ mJa )(n

Ka + iuK m

Ka )

(nIa + iuI mI

a)

∣∣∣∣∣ I 6= J 6= K 6= I (510)

Let us now turn to the fields in the fundamental representation of the gaugegroup Ga in particular to the fields arising from the intersection with one otherstack of branes denoted by b For an N = 1 open string sector the metric forthese fields can be written as [34 4] (see also [35])

Kabij = δij S

minusα

3prod

I=1

Uminus(β+ξ θI

ab)

I Tminus(γ+ζ θI

ab)

I

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (511)

where α β γ ξ and ζ are undetermined constants As θ12ab gt 0 and θ3ab lt 0which is assumed in (511) the intersection number Iab is positive implying that

nf = IabNb (512)

Using Ta(f) =12and relations (52 56 58 511 512) one finds a contribution

to the right hand side of (51) proportional to

IabNb

2

(ln

(M2

s

micro2

)+ (2γ minus 1) ln(T1T2T3) + (2β minus 1

2) ln(U1U2U3) + (2αminus 1

2) lnS

+ζ

3sum

I=1

θIab lnTI + ξ

3sum

I=1

θIab lnUI minus ln

[Γ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab)

]) (513)

Using (36) one finds that the first and the last term exactly reproduce the contri-bution of the last two terms in (31) The terms proportional to ζ and ξ will laterbe shown to constitute the aforementioned universal gauge coupling correctionThe remaining three terms can neither be attributed to such a correction nor canthey be written as the real part of a holomorphic function Thus they cannot bethe one-loop correction to the gauge kinetic function and therefore must vanishThis fixes some of the coefficients in the ansatz (511)

α = β =1

4 γ =

1

2 (514)

1An overall factor involving the wrapping numbers was introduced in this expression in orderto achieve full cancellation This can be done as the expressions used are derived only up tooverall constants [34]

12

The same matching of terms appears between the Mobius diagram plus the annu-lus with boundaries on brane a and its orientifold image and the Kahler metricsfor fields in the symmetric and antisymmetric representation Here one has toreplace θIab and IabNb by θIaaprime = 2θIa and IaaprimeNa in (511) and (33) Apart fromthese replacements the Kahler metric for matter in these representations is alsogiven by (511) with the constants α β γ given in (514)

The corrections to the gauge couplings coming fromN = 2 open string sectorswere seen in the previous section to take on quite a different form They containa term

minus ln |η(i T cI )|4 = minus4Re [ln η(i T c

I )] (515)

which can be written as the real part of a holomorphic function This leads oneto conclude that the gauge kinetic function receives one-loop corrections fromthese sectors Inserting the correct prefactor which from the first term in (35)and the corresponding one in (51) can be seen to be proportional to the betafunction coefficient gives

f (1)a = minusNb |IJab IKab |

4π2ln η(i T c

I ) I 6= J 6= K 6= I (516)

where again I denotes the torus in which the branes lie on top of each other andIJKab are the intersection numbers on the other tori

The term minus ln(TI VaI ) in (35) is not the real part of a holomorphic function

Proceeding as before one finds that the Kahler metric for the hypermultiplet (ortwo chiral multiplets) living at an intersection of branes a and b preserving eightsupercharges must be

KIij =

|nIa + iuI m

Ia|

(UJ UK T J TK)12

I 6= J 6= K 6= I (517)

Apart from the factor in the numerator this is in agreement with the form foundby direct calculations [34 4] The appearance of the numerator is however plau-sible as it also appears in the open string moduli metric and the hypermultipletsunder discussion should feel the Irsquoth torus in the same way

52 Universal threshold corrections

In the following the aforementioned ldquouniversalrdquo gauge coupling corrections willbe discussed They also appear in the heterotic [13] and type I [36 37] string andare related to a redefinition of the dilaton at one-loop [9 8] This stems from thefact that the dilaton really lives in a linear multiplet rather than a chiral one

Our general ansatz for the Kahler metrics for the chiral matter in an N = 1sector contains a factor

3prod

J=1

Uminusξ θJ

ab

J Tminusζ θJ

ab

J (518)

13

which according to (51) appears in the one-loop correction to the gauge couplingconstant Neither is this term reproduced in the string one-loop calculation ofthe coupling nor can it be written as a correction to the holomorphic gaugekinetic function Therefore as is familiar from gauge threshold computationsthere remains the possibility that it can be absorbed into a one-loop correctionto the S and UI chiral superfields In the following we require that such a gaugegroup factor independent universal correction is possible and see how this fixesthe parameters in (518)

The first observation is that in order to get something gauge group indepen-dent the factor (518) actually must have the following form

3prod

J=1

Uminusξprime sign(Iab)θ

Jab

J Tminusζprime sign(Iab)θ

Jab

J (519)

with ξprime and ζ prime independent of the brane For the metrics of fields transformingin the symmetric or antisymmetric representation of the gauge group one hasto replace φab = φa minus φb by φaaprime = 2φa and sign(Iab) by sign(Iaaprime minus IaO6) orsign(Iaaprime + IaO6) respectively Then one computes (K prime denotes the factor (518519) appearing in the full Kahler metric K)

sum

r

Ta(r) ln detKprimer = |Iab|Nb

2ln

[3prod

J=1

Uminusξprime sign(Iab) θ

Jab

J Tminusζprime sign(Iab) θ

Jab

J

](520)

+|Iabprime |Nb

2ln

[3prod

J=1

Uminusξprime sign(Iabprime) θ

Jabprime

J Tminusζprime sign(Iabprime ) θ

Jabprime

J

]

+ Na+22

|IaaprimeminusIaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(IaaprimeminusIaO6) θ

Ja

J Tminus2ζprime sign(IaaprimeminusIaO6) θ

Ja

J

]

+ Naminus22

|Iaaprime+IaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(Iaaprime+IaO6) θ

Ja

J Tminus2ζprime sign(Iaaprime+IaO6) θ

Ja

J

]

After a few steps using |Iab| sign(Iab) = Iab and the tadpole cancellation condi-tion this can be brought to the simple form

sum

r

Ta(r) ln detKprimer = minusn1

an2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

minus3sum

J=1

nJam

Ka m

La

[sum

b

NbmJb n

Kb n

Lb

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

J 6= K 6= L 6= J (521)

Therefore these corrections have precisely the form required for them to be iden-tified with the one-loop correction between the linear superfields appearing in

14

string theory and the chiral superfields used in the supergravity description

SL = S minus 1

8π2

sum

b

Nbm1bm

2bm

3b

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

ULJ = UJ +

1

8π2

sum

b

NbmJb n

Kb n

Lb

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

J 6= K 6= L 6= J (522)

In contrast to eq (59) where the tree-level gauge couplings are determined theone-loop gauge couplings at the string scale have to include the redefined fieldsSL and UL

gminus2astring = SL

3prod

I=1

nIa minus

3sum

I=1

ULI n

Iam

Jam

Ka (523)

In contrast to all (to us) known cases studied in the literature for ξprime 6= 0 thefields which are corrected ie the moduli S and UI also appear in the one-loopredefinition Let us propose an argument why such corrections might be expectedto be absent Due to the anomalous U(1) gauge symmetries the chiral superfieldsS and UI participate in the Green-Schwarz mechanism and therefore transformnon-trivially under U(1) gauge transformations This implies that in order tobe gauge invariant the one-loop corrections in (522) proportional to lnUI mustbe extended in the usual way by UI rarr UI + δaGSVa Computing the resultingFI-terms via the supergravity formula ξa2g

2a = partKpartVa|Va=0 gives besides the

tree-level result depending on SL ULI a one-loop contribution proportional to

ξprimesum

b ξ(0)b φJ

b UJ This has an extra dependence on the complex structure moduliUI However for intersecting D6-branes we have seen that the Wilsonian (super-gravity) FI-Terms are proportional to the Wilsonian gauge threshold correctionswhich depend only on the Kahler moduli via instanton corrections (for N = 1sectors they are even vanishing in the setup at hand) This seems to suggest thatthere should better be no UminusθI

I dependence in the matter field Kahler metricsie ξprime = 0

Moreover in analogy to the heterotic string we expect that for the rangeminus1 le θJ le 1 the exponent of TJ runs over the range [minus1 0] This conditionwould fix ζ prime = plusmn12

Let us summarise the conclusions we have drawn from requiring holomorphyof the Wilsonian gauge kinetic function First we provided arguments that theKahler metric for N = 1 chiral matter fields in intersecting D6-brane models isof the following form

Kabij = δij S

minus 14

3prod

J=1

Uminus 1

4J T

minus( 12plusmn 1

2sign(Iab) θ

Jab)

J

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (524)

15

where supersymmetry of course requiressum3

I=1 θIab = 0 Second the holomorphic

gauge kinetic function (on the background considered) only receives correctionsfrom N = 2 open string sectors and the one-loop correction takes on the followingform

f (1)a = minus

sum

b

Nb |IJabIKab |4π2

ln η(i T cI ) I 6= J 6= K 6= I (525)

where the sum only runs over branes b which lie on top of brane a in exactly onetorus denoted by I Therefore the results for the gauge threshold corrections andthe matter field Kahler metrics are consistent both with the non-renormalisationtheorem from section 2 and the Kaplunovsky-Louis formula (51) Clearly itwould be interesting along the lines of [34] to carry out a string amplitude com-putation to fix the free coefficient in the ansatz (511) and see whether our indirectarguments are correct

6 Holomorphic E2-instanton amplitudes

Space-time instantons are given also by D-branes which in this case are EuclideanD2-branes (so-called E2-branes) wrapping three-cycles Ξ in the CalabindashYau sothat they are point-like in four-dimensional Minkowski space Such instantonscan contribute to the holomorphic superpotential and gauge kinetic functions onlyif they preserve half of the N = 1 supersymmetry This means that the instantonmeasure must contain a factor d4x d2θ Let us first clarify an important aspectof this half-BPS condition In the second part of this section we then revisitthe computation of contributions of such instantons to the superpotential andalso clarify some issues concerned with the appearing one-loop determinants Inthe third and fourth part we investigate under which conditions such stringinstantons can also contribute to the gauge kinetic functions and FI-terms

61 Half-BPS instantons

As has been explained in [24 18 20] just wrapping an E2-instanton around arigid sLag three-cycle in the Calabi-Yau gives four bosonic and four fermioniczero modes The vertex operators for the latter are

V(minus12)θ (z) = θα e

minusϕ(z)2 (z)Sα(z) Σh= 3

8q= 3

2(z) (61)

and

V(minus12)

θ(z) = θα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 3

2(z) (62)

Therefore if the instanton is not invariant under the orientifold projection onestill has four instead of the desired two fermionic zero modes Thus only by

16

placing the E2-brane in a position invariant under Ωσ does one have a chanceto get rid of the two additional zero modes θ For so called O(n) instantons onecan see that the zero modes xmicro θ are symmetrised and the mode θ gets anti-symmetrised For the opposite projection ie for USp(2n) instantons the zeromodes xmicro θ are anti-symmetrised and the mode θ gets symmetrised Thereforeone can only get the simple d4x d2θ instanton measure for a single O(1) instanton

62 Superpotential contributions

In order to contribute to the superpotential we also require that there do notarise any further zero modes from E2-E2 open strings so that the three-cycleΞ should be rigid ie b1(Ξ) = 0 Therefore considering an E2-instanton in anintersecting brane configuration additional zero modes can only arise from theintersection of the instanton Ξ with D6-branes Πa There are Na [ΞcapΠa]

+ chiralfermionic zero modes λaI and Na [Ξ cap Πa]

minus anti-chiral ones λaJ 2

For its presentation it is useful to introduce the short-hand notation

Φak bk [~xk] = Φak xk1middot Φxk1xk2

middot Φxk2xk3middot middot Φxknminus1xkn

middot Φxkn(k)bk (63)

for the chain-product of open string vertex operators Here we define Φak bk [~0] =Φak bk

To extract the superpotential one can probe it by evaluating an appropriatematter field correlator in the instanton background The CFT allows one tocompute it in physical normalisation which combines the superpotential part Ywith the matter field Kahler metrics like

〈Φa1b1 middot middot ΦaM bM 〉E2minusinst =e

K

2 YΦa1b1ΦaMbMradic

Ka1b1 middot middotKaM bM

(64)

In [14] a general expression for the single E2-instanton contribution to thecharged matter superpotential was proposed involving the evaluation of the fol-lowing zero mode integral over disc and one-loop open string CFT amplitudes

〈Φa1b1 middot middot ΦaM bM 〉E2 =V3gs

intd4x d2θ

sum

conf

proda

(prod[ΞcapΠa]+

i=1 dλia) (prod[ΞcapΠa]minus

i=1 dλi

a

)

exp(minusSE2) exp (Z prime0(E2)) 〈Φa1b1[~x1]〉λa1 λb1

middot middot 〈ΦaLbL[~xL]〉λaLλbL

(65)

For simplicity we do not consider the case that matter fields are also assignedto string loop diagrams The one-loop contributions are annulus diagrams for

2Here we introduced the physical intersection number between two branes Πa cap Πb whichis the sum of positive [Πa capΠb]

+ and negative [Πa cap Πb]minus intersections

17

open strings with one boundary on the E2-instanton and the other boundary onthe various D6-branes and Mobius diagrams with boundary on the E2-instanton

〈1〉1-loop = Z prime0(E2) =

sumb Z

primeA(E2aD6b) + Z primeM(E2aO6) (66)

Here Z prime means that we only sum over the massive open string states in theloop amplitude as the zero modes are taken care of explicitly It was shownthat these instantonic open string loop diagrams are identical to the one-loopthreshold corrections TA(D6aD6b) Diagrammatically we have the intriguingrelation shown in figure 1 and in figure 2 which holds for the even spin structures3

Fa

Fa x

x

D6aaE2 =D6b D6b

Figure 1 Relation between instantonic one-loop amplitudes and corresponding gaugethreshold corrections

Fa

Fa x

x

D6aaE2 =

O6O6

Figure 2 Relation between instantonic Mobius amplitude and corresponding gaugethreshold corrections

The annulus threshold corrections can be computed leading to

ZA(E2aD6b) =

int infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it)

η3(it)ACY

ab [αβ](it) (67)

and the Mobius strip amplitude for the instanton which as we explained mustbe invariant under the orientifold projection yields

ZM(E2aO6) = plusmnint infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it + 1

2

)

η3(it + 1

2

) ACYaa [α

β](it + 1

2

)(68)

3The contribution of the CP-odd Rminus sector is expected to yield corrections to the θ-angle

18

The overall plus sign is for O(1) instantons reflecting the fact that only for thesethe xmicro and θα zero modes survive the orientifold projection Note that up to theargument the Mobius thresholds are ZA(E2aD6a) Therefore for rigid branesthe massless sector reflects the number of four bosonic and two fermionic zeromodes In section 63 we will discuss the number of zero modes if b1(Ξ) gt 0 Allthese stringy threshold corrections are known to be non-holomorphic Thereforeit is not immediately obvious in which sense the expression (65) is meant andhow one can extract the holomorphic superpotential part Y from it

The CFT disc amplitudes in (65) are also not holomorphic but combinenon-holomorphic Kahler potential contributions and holomorphic superpotentialcontributions in the usual way [38 39 40 41]

〈Φab[~x]〉λaλb=

eK

2 YλaΦax1Φx1x2 ΦxN b λbradicKλaaKax1 KxnbKbλb

(69)

=e

K

2 YλabΦab[x]λbradic

Kλaa Kab[x]Kbλb

(610)

Due to the Kaplunovsky-Louis formula (51) the stringy one-loop amplitudes areknown to include the holomorphic Wilsonian part and contributions from wave-function normalisation Applied to the instanton one-loop amplitudes appearingin Z0(E2a) we write

Z0(E2a) = minus8π2 real(f (1)a )minus ba

2ln

(M2

p

micro2

)minus ca

2Ktree (611)

minus ln

(V3gs

)

tree

+sum

b

|IabNb|2

ln[detKab

]tree

where for the brane and instanton configuration in question the coefficients are

ba =sum

b

|IabNb|2

minus 3 ca =sum

b

|IabNb|2

minus 1 (612)

The constant contributions arise from the Mobius amplitude Inserting (69) and(611) in (65) one realises that the Kahler metrics involving an instanton zeromode and a matter field precisely cancel out so that only the matter metricssurvive as required by the general form (64) Moreover the term exp(K2)comes out just right due to the rule that each disc contains precisely two instantonzero modes The holomorphic piece in (64) can therefore be expressed entirelyin terms of other holomorphic quantities like holomorphic Yukawa couplingsthe holomorphic instanton action and the one-loop holomorphic Wilsonian gaugekinetic function on the E2-brane

YΦa1b1ΦaMbM

=sum

conf

signconf exp(minusSE2)tree exp(minusf (1)

a

)

Yλa1bΦa1b1

[~x1] λb1middot middot Yλa1

bΦaLbL[~xL]λbL

(613)

19

This explicitly shows that knowing the tree-level Kahler potentials computingthe matter field correlator in the instanton background up to one-loop level ings is sufficient to deduce the Wilsonian holomorphic instanton generated super-potential Higher order corrections in gs only come from loop corrections to theKahler potentials

63 Instanton corrections to the gauge kinetic functions

So far we have discussed space-time instanton corrections to the superpotentialThese involved one-loop determinants which are given by annulus vacuum di-agrams with at least one E2 boundary These are related to one-loop gaugethreshold corrections to the gauge theory on a D6-brane wrapping the same cycleas the E2 instanton

Now we can ask what other corrections these space-time instantons can induceBy applying S- and T-dualities to the story of world-sheet instanton correctionsin the heterotic string we expect that there can also be E2-instanton corrections

to the holomorphic gauge kinetic functions In the heterotic case similar to thetopological Type II string such corrections arise from string world-sheets of Eulercharacteristic zero ie here from world-sheets with two boundaries Thereforewe expect such corrections to appear for E2-instantons admitting one complexopen string modulus ie those wrapping a three-cycle with Betti number b1(Ξ) =1

Let us start by discussing the instanton zero mode structure for such a cycleFirst let us provide the form of the vertex operators The bosonic fields in the(minus1) ghost picture are

V (minus1)y (z) = y eminusϕ(z)Σh= 1

2q=plusmn1(z) (614)

which before the orientifold projection are accompanied by the two pairs offermionic zero modes

V (minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 1

2(z) (615)

and

V(minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=+ 1

2(z) (616)

Now one has to distinguish two cases depending on how the anti-holomorphicinvolution σ acts on the open string modulus Y

σ y rarr plusmny (617)

In the case that y is invariant under σ called first kind in the following theorientifold projection acts in the same way as for the 4D fields Xmicro ie thetwo bosonic zero modes y and the two fermionic zero modes micro survive In the

20

other case dubbed second kind the bosonic zero mode is projected out andonly the fermionic modulino zero mode micro survives4 Therefore in the absence ofany additional zero modes for instance from E2-D6 intersections the zero modemeasure in any instanton amplitude assumes the following form

intd4x d2θ d2y d2micro eminusSE2 for σ y rarr y (618)

andintd4x d2θ d2micro eminusSE2 for σ y rarr minusy (619)

As an example consider the set-up in figure 3 with σ yi rarr minusyi Here the

x x x2 31

1 2 3y y y

O6

E2

∆x1 ∆ y3

∆x2

Figure 3 Deformations of an instanton which is invariant under the orientifold pro-jection

deformations ∆x12 are of the first kind and ∆y3 is of the second kindNow it is clear that an instanton with precisely one set of fermionic zero

modes of the second kind and no additional zero modes can generate a correctionto the SU(Na) gauge kinetic function The instanton amplitude takes on thefollowing form

〈Fa(p1)Fa(p2)〉E2 =

intd4x d2θ d2micro exp(minusSE2) exp (Z prime

0(E2)) AF 2a(E2 D6a)

where AF 2a(E2 D6a) is the annulus diagram in figure 4 which absorbs all the

appearing fermionic zero modes and where the gauge boson vertex operators inthe (0)-ghost picture have the usual form

V(0)A (z) = ǫmicro (partmicroX(z) + i(p middot ψ)ψmicro(z)) e

ipmiddotX(z) (620)

4By duality this distinction is related to the two kinds of deformations of genus g curvesstudied in [42] The first kind are the curves moving in families ie transversal deformationof the curve The second kind is related to the deformations coming with the genus g of thecurve

21

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

The derivative with respect to the supersymmetry breaking parameters ǫI bringsdown one factor of t and it turns out that the result is the same for all partǫI

〈VDa〉ǫ ≃ i IabNb

int infin

0

dt

t

3sum

I=1

ϑprime1ϑ1

(iθI

abt

2 it2

) (45)

This is the same expression as the one-loop threshold corrections TA(D6a D6b)so that at linear order in ǫ we obtain for the FI-terms

(ξa)ǫ = (ǫb minus ǫa) TA(D6a D6b) (46)

Completely analogously one can show that this formula remains true also forN = 2 open string sectors Therefore we would like to propose that such arelation between gauge threshold and one-loop corrections to FI-terms is validfor general intersecting D6-brane models Moreover the Wilsonian part of thethresholds TA(D6a D6b) which we compute in the next section should also bethe Wilsonian part of the correction to the FI-terms

5 Wilsonian gauge kinetic function and σ-model

anomalies

In a supersymmetric gauge theory one can compute the running gauge couplingsga(micro

2) in terms of the gauge kinetic functions fa the Kahler potential K and theKahler metrics of the charged matter fields Kab(micro2) [33 8 9 10 11 12 13]

8π2

g2a(micro2)

= 8π2real(fa) +ba2

ln

(Λ2

micro2

)+ca2K

+T (Ga) ln gminus2a (micro2)minus

sum

r

Ta(r) ln detKr(micro2) (51)

with

ba =sum

r

nrTa(r)minus 3 T (Ga) ca =sum

r

nrTa(r)minus T (Ga) (52)

and Ta(r) = Tr(T 2(a)) (T(a) being the generators of the gauge group Ga) In addi-

tion T (Ga) = Ta(adjGa) and nr is the number of multiplets in the representationr of the gauge group and the sums run over these representations In this contextthe natural cutoff scale for a field theory is the Planck scale ie Λ2 =M2

PlThe left hand side of eq (51) is given by eq (31) which contains the gauge

coupling at the string scale 1g2astring as well as the one-loop string thresholdcorrections ∆a In general ∆a is the sum of a non-holomorphic term plus thereal-part of a holomorphic threshold correction

∆a = ∆nha + real(∆hol

a ) (53)

9

On the right hand side of eq (51) fa denotes the Wilsonian ie holomorphicgauge kinetic function which is given in terms of a holomorphic tree-level functionplus the holomorphic part of the one-loop threshold corrections (cf eq (27))

fa = f treea + f 1minusloop

a

(eminusT c

i

)=sum

I

M IaU

cI +∆hol

a (54)

In addition on the right hand side of eq (51) the non-holomorphic terms propor-tional to the Kahler metric of the moduli K and the matter field Kahler metricsKr are due to the one-loop contributions of massless fields These fields generatenon-local terms in the one-loop effective action which correspond to one-loopnon-invariances under σ-model transformations the so-called σ-model anomalies(Kahler and reparametrisation anomalies)

Matching up all terms in eq (51) essentially means that the σ-model anoma-lies can be cancelled in a two-fold way First by local contributions to the gaugecoupling constant via the one-loop threshold contributions ∆a These terms origi-nate from massive string states The second way to cancel the σ-model anomaliesis due to a field dependent (however gauge group independent) one-loop contribu-tion to the Kahler potential of the chiral moduli fields It implies that some of themoduli fields transform non-trivially under the Kahler transformations and alsounder reparametrisations in moduli space The universal one-loop modificationof the Kahler potential is nothing else than a generalised Green-Schwarz mecha-nism cancelling the σ-model anomalies This is analogous to the Green-Schwarzmechanism which cancels anomalies of physical U(1) gauge fields whereas theσ-model anomalies correspond to unphysical composite gauge connections Ef-fectively it means that the Green-Schwarz mechanism with respect to the σ-modelanomalies can be described by a non-holomorphic one-loop field redefinition ofthe associated tree-level moduli fields

As we will see in type IIA orientifold models these field redefinitions act onthe real parts of the dilaton field S as well as the complex structure moduli UJ

S rarr S + δGS(U T )

UJ rarr UJ + δGSJ (U T ) (55)

These redefined fields are those that determine the gauge coupling constants1g2astring at the string scale Recall that as the tree-level gauge couplings (24)are already gauge group dependent so are these one-loop corrections but theonly dependence arises due to the universal one-loop redefinition of the modulifields It is in this sense that we still call these one-loop corrections to the gaugecouplings universal Note that in heterotic string compactifications the σ-modelGreen-Schwarz mechanism only acts on the heterotic dilaton field

10

51 Holomorphic gauge couplings for toroidal models

In summary equation (51) is to be understood recursively which means that onecan insert the tree-level results into the last three terms of eq (51) In additionone also has to include the universal field redefinition eq (55) in 1g2astring inthe left hand side of eq (51) in order to get a complete matching of all termsin eq (51) as we will demonstrate for the aforementioned toroidal orbifold inthe following For N = 2 sectors the one-loop threshold corrections to the gaugecoupling constant indeed contain a holomorphic Wilsonian term f

(1)a whereas

for N = 1 sectors only the non-holomorphic piece ∆nha is present

Specifically the holomorphic gauge kinetic function can now be determinedby comparing the string theoretical formula (31) for the effective gauge couplingwith the field theoretical one (51) The first thing to notice are the differentcutoff scales appearing in the two formulas One needs to convert one into theother using

M2s

M2Pl

prop exp(2φ4) prop (S U1 U2 U3)minus 1

2 (56)

Here φ4 is the four dimensional dilaton and the complex structure moduli in thesupergravity basis can be expressed in terms of φ4 and the complex structuremoduli uI = RI2RI1 as

S =1

2πeminusφ4

1radicu1 u2 u3

UI =1

2πeminusφ4

radicuJ uKuI

with I 6= J 6= K 6= I (57)

These fields are the real parts of complex scalars of four dimensional chiral mul-tiplets Sc and U c

I As N = 4 super YangndashMills theory is finite one expects the sum of the terms

in (31) proportional to T (Ga) to cancel This is because the only chiral multipletstransforming in the adjoint representation of the gauge group are the open stringmoduli which (on the background considered) assemble themselves into threechiral multiplets thus forming an N = 4 sector together with the gauge fieldsTo show that this cancellation does happen one notices the following Firstlynadjoint = 3 as explained such that there is no term in ba proportional to T (Ga)Secondly

K = minus ln(Sc + Sc)minus

3sum

I=1

ln(U cI + U

c

I)minus3sum

I=1

ln(T cI + T

c

I) (58)

gminus2atree = S

3prod

I=1

nIa minus

3sum

I=1

UI nIam

Jam

Ka I 6= J 6= K 6= I (59)

where TI are the Kahler moduli of the torus and nIa m

Ia are the wrapping numbers

of the brane Finally one needs the matter metric for the open string moduli

11

which can be obtained from the T-dual expression in models with D9- and D5-branes [4] Performing the T-duality which essentially amounts to exchangingKahler and complex structure moduli and converting gauge flux into non-trivialintersection angles for the D6-branes one arrives at (I = 1 3)1

KIij =

δijTIUI

∣∣∣∣∣(nJ

a + iuJ mJa )(n

Ka + iuK m

Ka )

(nIa + iuI mI

a)

∣∣∣∣∣ I 6= J 6= K 6= I (510)

Let us now turn to the fields in the fundamental representation of the gaugegroup Ga in particular to the fields arising from the intersection with one otherstack of branes denoted by b For an N = 1 open string sector the metric forthese fields can be written as [34 4] (see also [35])

Kabij = δij S

minusα

3prod

I=1

Uminus(β+ξ θI

ab)

I Tminus(γ+ζ θI

ab)

I

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (511)

where α β γ ξ and ζ are undetermined constants As θ12ab gt 0 and θ3ab lt 0which is assumed in (511) the intersection number Iab is positive implying that

nf = IabNb (512)

Using Ta(f) =12and relations (52 56 58 511 512) one finds a contribution

to the right hand side of (51) proportional to

IabNb

2

(ln

(M2

s

micro2

)+ (2γ minus 1) ln(T1T2T3) + (2β minus 1

2) ln(U1U2U3) + (2αminus 1

2) lnS

+ζ

3sum

I=1

θIab lnTI + ξ

3sum

I=1

θIab lnUI minus ln

[Γ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab)

]) (513)

Using (36) one finds that the first and the last term exactly reproduce the contri-bution of the last two terms in (31) The terms proportional to ζ and ξ will laterbe shown to constitute the aforementioned universal gauge coupling correctionThe remaining three terms can neither be attributed to such a correction nor canthey be written as the real part of a holomorphic function Thus they cannot bethe one-loop correction to the gauge kinetic function and therefore must vanishThis fixes some of the coefficients in the ansatz (511)

α = β =1

4 γ =

1

2 (514)

1An overall factor involving the wrapping numbers was introduced in this expression in orderto achieve full cancellation This can be done as the expressions used are derived only up tooverall constants [34]

12

The same matching of terms appears between the Mobius diagram plus the annu-lus with boundaries on brane a and its orientifold image and the Kahler metricsfor fields in the symmetric and antisymmetric representation Here one has toreplace θIab and IabNb by θIaaprime = 2θIa and IaaprimeNa in (511) and (33) Apart fromthese replacements the Kahler metric for matter in these representations is alsogiven by (511) with the constants α β γ given in (514)

The corrections to the gauge couplings coming fromN = 2 open string sectorswere seen in the previous section to take on quite a different form They containa term

minus ln |η(i T cI )|4 = minus4Re [ln η(i T c

I )] (515)

which can be written as the real part of a holomorphic function This leads oneto conclude that the gauge kinetic function receives one-loop corrections fromthese sectors Inserting the correct prefactor which from the first term in (35)and the corresponding one in (51) can be seen to be proportional to the betafunction coefficient gives

f (1)a = minusNb |IJab IKab |

4π2ln η(i T c

I ) I 6= J 6= K 6= I (516)

where again I denotes the torus in which the branes lie on top of each other andIJKab are the intersection numbers on the other tori

The term minus ln(TI VaI ) in (35) is not the real part of a holomorphic function

Proceeding as before one finds that the Kahler metric for the hypermultiplet (ortwo chiral multiplets) living at an intersection of branes a and b preserving eightsupercharges must be

KIij =

|nIa + iuI m

Ia|

(UJ UK T J TK)12

I 6= J 6= K 6= I (517)

Apart from the factor in the numerator this is in agreement with the form foundby direct calculations [34 4] The appearance of the numerator is however plau-sible as it also appears in the open string moduli metric and the hypermultipletsunder discussion should feel the Irsquoth torus in the same way

52 Universal threshold corrections

In the following the aforementioned ldquouniversalrdquo gauge coupling corrections willbe discussed They also appear in the heterotic [13] and type I [36 37] string andare related to a redefinition of the dilaton at one-loop [9 8] This stems from thefact that the dilaton really lives in a linear multiplet rather than a chiral one

Our general ansatz for the Kahler metrics for the chiral matter in an N = 1sector contains a factor

3prod

J=1

Uminusξ θJ

ab

J Tminusζ θJ

ab

J (518)

13

which according to (51) appears in the one-loop correction to the gauge couplingconstant Neither is this term reproduced in the string one-loop calculation ofthe coupling nor can it be written as a correction to the holomorphic gaugekinetic function Therefore as is familiar from gauge threshold computationsthere remains the possibility that it can be absorbed into a one-loop correctionto the S and UI chiral superfields In the following we require that such a gaugegroup factor independent universal correction is possible and see how this fixesthe parameters in (518)

The first observation is that in order to get something gauge group indepen-dent the factor (518) actually must have the following form

3prod

J=1

Uminusξprime sign(Iab)θ

Jab

J Tminusζprime sign(Iab)θ

Jab

J (519)

with ξprime and ζ prime independent of the brane For the metrics of fields transformingin the symmetric or antisymmetric representation of the gauge group one hasto replace φab = φa minus φb by φaaprime = 2φa and sign(Iab) by sign(Iaaprime minus IaO6) orsign(Iaaprime + IaO6) respectively Then one computes (K prime denotes the factor (518519) appearing in the full Kahler metric K)

sum

r

Ta(r) ln detKprimer = |Iab|Nb

2ln

[3prod

J=1

Uminusξprime sign(Iab) θ

Jab

J Tminusζprime sign(Iab) θ

Jab

J

](520)

+|Iabprime |Nb

2ln

[3prod

J=1

Uminusξprime sign(Iabprime) θ

Jabprime

J Tminusζprime sign(Iabprime ) θ

Jabprime

J

]

+ Na+22

|IaaprimeminusIaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(IaaprimeminusIaO6) θ

Ja

J Tminus2ζprime sign(IaaprimeminusIaO6) θ

Ja

J

]

+ Naminus22

|Iaaprime+IaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(Iaaprime+IaO6) θ

Ja

J Tminus2ζprime sign(Iaaprime+IaO6) θ

Ja

J

]

After a few steps using |Iab| sign(Iab) = Iab and the tadpole cancellation condi-tion this can be brought to the simple form

sum

r

Ta(r) ln detKprimer = minusn1

an2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

minus3sum

J=1

nJam

Ka m

La

[sum

b

NbmJb n

Kb n

Lb

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

J 6= K 6= L 6= J (521)

Therefore these corrections have precisely the form required for them to be iden-tified with the one-loop correction between the linear superfields appearing in

14

string theory and the chiral superfields used in the supergravity description

SL = S minus 1

8π2

sum

b

Nbm1bm

2bm

3b

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

ULJ = UJ +

1

8π2

sum

b

NbmJb n

Kb n

Lb

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

J 6= K 6= L 6= J (522)

In contrast to eq (59) where the tree-level gauge couplings are determined theone-loop gauge couplings at the string scale have to include the redefined fieldsSL and UL

gminus2astring = SL

3prod

I=1

nIa minus

3sum

I=1

ULI n

Iam

Jam

Ka (523)

In contrast to all (to us) known cases studied in the literature for ξprime 6= 0 thefields which are corrected ie the moduli S and UI also appear in the one-loopredefinition Let us propose an argument why such corrections might be expectedto be absent Due to the anomalous U(1) gauge symmetries the chiral superfieldsS and UI participate in the Green-Schwarz mechanism and therefore transformnon-trivially under U(1) gauge transformations This implies that in order tobe gauge invariant the one-loop corrections in (522) proportional to lnUI mustbe extended in the usual way by UI rarr UI + δaGSVa Computing the resultingFI-terms via the supergravity formula ξa2g

2a = partKpartVa|Va=0 gives besides the

tree-level result depending on SL ULI a one-loop contribution proportional to

ξprimesum

b ξ(0)b φJ

b UJ This has an extra dependence on the complex structure moduliUI However for intersecting D6-branes we have seen that the Wilsonian (super-gravity) FI-Terms are proportional to the Wilsonian gauge threshold correctionswhich depend only on the Kahler moduli via instanton corrections (for N = 1sectors they are even vanishing in the setup at hand) This seems to suggest thatthere should better be no UminusθI

I dependence in the matter field Kahler metricsie ξprime = 0

Moreover in analogy to the heterotic string we expect that for the rangeminus1 le θJ le 1 the exponent of TJ runs over the range [minus1 0] This conditionwould fix ζ prime = plusmn12

Let us summarise the conclusions we have drawn from requiring holomorphyof the Wilsonian gauge kinetic function First we provided arguments that theKahler metric for N = 1 chiral matter fields in intersecting D6-brane models isof the following form

Kabij = δij S

minus 14

3prod

J=1

Uminus 1

4J T

minus( 12plusmn 1

2sign(Iab) θ

Jab)

J

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (524)

15

where supersymmetry of course requiressum3

I=1 θIab = 0 Second the holomorphic

gauge kinetic function (on the background considered) only receives correctionsfrom N = 2 open string sectors and the one-loop correction takes on the followingform

f (1)a = minus

sum

b

Nb |IJabIKab |4π2

ln η(i T cI ) I 6= J 6= K 6= I (525)

where the sum only runs over branes b which lie on top of brane a in exactly onetorus denoted by I Therefore the results for the gauge threshold corrections andthe matter field Kahler metrics are consistent both with the non-renormalisationtheorem from section 2 and the Kaplunovsky-Louis formula (51) Clearly itwould be interesting along the lines of [34] to carry out a string amplitude com-putation to fix the free coefficient in the ansatz (511) and see whether our indirectarguments are correct

6 Holomorphic E2-instanton amplitudes

Space-time instantons are given also by D-branes which in this case are EuclideanD2-branes (so-called E2-branes) wrapping three-cycles Ξ in the CalabindashYau sothat they are point-like in four-dimensional Minkowski space Such instantonscan contribute to the holomorphic superpotential and gauge kinetic functions onlyif they preserve half of the N = 1 supersymmetry This means that the instantonmeasure must contain a factor d4x d2θ Let us first clarify an important aspectof this half-BPS condition In the second part of this section we then revisitthe computation of contributions of such instantons to the superpotential andalso clarify some issues concerned with the appearing one-loop determinants Inthe third and fourth part we investigate under which conditions such stringinstantons can also contribute to the gauge kinetic functions and FI-terms

61 Half-BPS instantons

As has been explained in [24 18 20] just wrapping an E2-instanton around arigid sLag three-cycle in the Calabi-Yau gives four bosonic and four fermioniczero modes The vertex operators for the latter are

V(minus12)θ (z) = θα e

minusϕ(z)2 (z)Sα(z) Σh= 3

8q= 3

2(z) (61)

and

V(minus12)

θ(z) = θα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 3

2(z) (62)

Therefore if the instanton is not invariant under the orientifold projection onestill has four instead of the desired two fermionic zero modes Thus only by

16

placing the E2-brane in a position invariant under Ωσ does one have a chanceto get rid of the two additional zero modes θ For so called O(n) instantons onecan see that the zero modes xmicro θ are symmetrised and the mode θ gets anti-symmetrised For the opposite projection ie for USp(2n) instantons the zeromodes xmicro θ are anti-symmetrised and the mode θ gets symmetrised Thereforeone can only get the simple d4x d2θ instanton measure for a single O(1) instanton

62 Superpotential contributions

In order to contribute to the superpotential we also require that there do notarise any further zero modes from E2-E2 open strings so that the three-cycleΞ should be rigid ie b1(Ξ) = 0 Therefore considering an E2-instanton in anintersecting brane configuration additional zero modes can only arise from theintersection of the instanton Ξ with D6-branes Πa There are Na [ΞcapΠa]

+ chiralfermionic zero modes λaI and Na [Ξ cap Πa]

minus anti-chiral ones λaJ 2

For its presentation it is useful to introduce the short-hand notation

Φak bk [~xk] = Φak xk1middot Φxk1xk2

middot Φxk2xk3middot middot Φxknminus1xkn

middot Φxkn(k)bk (63)

for the chain-product of open string vertex operators Here we define Φak bk [~0] =Φak bk

To extract the superpotential one can probe it by evaluating an appropriatematter field correlator in the instanton background The CFT allows one tocompute it in physical normalisation which combines the superpotential part Ywith the matter field Kahler metrics like

〈Φa1b1 middot middot ΦaM bM 〉E2minusinst =e

K

2 YΦa1b1ΦaMbMradic

Ka1b1 middot middotKaM bM

(64)

In [14] a general expression for the single E2-instanton contribution to thecharged matter superpotential was proposed involving the evaluation of the fol-lowing zero mode integral over disc and one-loop open string CFT amplitudes

〈Φa1b1 middot middot ΦaM bM 〉E2 =V3gs

intd4x d2θ

sum

conf

proda

(prod[ΞcapΠa]+

i=1 dλia) (prod[ΞcapΠa]minus

i=1 dλi

a

)

exp(minusSE2) exp (Z prime0(E2)) 〈Φa1b1[~x1]〉λa1 λb1

middot middot 〈ΦaLbL[~xL]〉λaLλbL

(65)

For simplicity we do not consider the case that matter fields are also assignedto string loop diagrams The one-loop contributions are annulus diagrams for

2Here we introduced the physical intersection number between two branes Πa cap Πb whichis the sum of positive [Πa capΠb]

+ and negative [Πa cap Πb]minus intersections

17

open strings with one boundary on the E2-instanton and the other boundary onthe various D6-branes and Mobius diagrams with boundary on the E2-instanton

〈1〉1-loop = Z prime0(E2) =

sumb Z

primeA(E2aD6b) + Z primeM(E2aO6) (66)

Here Z prime means that we only sum over the massive open string states in theloop amplitude as the zero modes are taken care of explicitly It was shownthat these instantonic open string loop diagrams are identical to the one-loopthreshold corrections TA(D6aD6b) Diagrammatically we have the intriguingrelation shown in figure 1 and in figure 2 which holds for the even spin structures3

Fa

Fa x

x

D6aaE2 =D6b D6b

Figure 1 Relation between instantonic one-loop amplitudes and corresponding gaugethreshold corrections

Fa

Fa x

x

D6aaE2 =

O6O6

Figure 2 Relation between instantonic Mobius amplitude and corresponding gaugethreshold corrections

The annulus threshold corrections can be computed leading to

ZA(E2aD6b) =

int infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it)

η3(it)ACY

ab [αβ](it) (67)

and the Mobius strip amplitude for the instanton which as we explained mustbe invariant under the orientifold projection yields

ZM(E2aO6) = plusmnint infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it + 1

2

)

η3(it + 1

2

) ACYaa [α

β](it + 1

2

)(68)

3The contribution of the CP-odd Rminus sector is expected to yield corrections to the θ-angle

18

The overall plus sign is for O(1) instantons reflecting the fact that only for thesethe xmicro and θα zero modes survive the orientifold projection Note that up to theargument the Mobius thresholds are ZA(E2aD6a) Therefore for rigid branesthe massless sector reflects the number of four bosonic and two fermionic zeromodes In section 63 we will discuss the number of zero modes if b1(Ξ) gt 0 Allthese stringy threshold corrections are known to be non-holomorphic Thereforeit is not immediately obvious in which sense the expression (65) is meant andhow one can extract the holomorphic superpotential part Y from it

The CFT disc amplitudes in (65) are also not holomorphic but combinenon-holomorphic Kahler potential contributions and holomorphic superpotentialcontributions in the usual way [38 39 40 41]

〈Φab[~x]〉λaλb=

eK

2 YλaΦax1Φx1x2 ΦxN b λbradicKλaaKax1 KxnbKbλb

(69)

=e

K

2 YλabΦab[x]λbradic

Kλaa Kab[x]Kbλb

(610)

Due to the Kaplunovsky-Louis formula (51) the stringy one-loop amplitudes areknown to include the holomorphic Wilsonian part and contributions from wave-function normalisation Applied to the instanton one-loop amplitudes appearingin Z0(E2a) we write

Z0(E2a) = minus8π2 real(f (1)a )minus ba

2ln

(M2

p

micro2

)minus ca

2Ktree (611)

minus ln

(V3gs

)

tree

+sum

b

|IabNb|2

ln[detKab

]tree

where for the brane and instanton configuration in question the coefficients are

ba =sum

b

|IabNb|2

minus 3 ca =sum

b

|IabNb|2

minus 1 (612)

The constant contributions arise from the Mobius amplitude Inserting (69) and(611) in (65) one realises that the Kahler metrics involving an instanton zeromode and a matter field precisely cancel out so that only the matter metricssurvive as required by the general form (64) Moreover the term exp(K2)comes out just right due to the rule that each disc contains precisely two instantonzero modes The holomorphic piece in (64) can therefore be expressed entirelyin terms of other holomorphic quantities like holomorphic Yukawa couplingsthe holomorphic instanton action and the one-loop holomorphic Wilsonian gaugekinetic function on the E2-brane

YΦa1b1ΦaMbM

=sum

conf

signconf exp(minusSE2)tree exp(minusf (1)

a

)

Yλa1bΦa1b1

[~x1] λb1middot middot Yλa1

bΦaLbL[~xL]λbL

(613)

19

This explicitly shows that knowing the tree-level Kahler potentials computingthe matter field correlator in the instanton background up to one-loop level ings is sufficient to deduce the Wilsonian holomorphic instanton generated super-potential Higher order corrections in gs only come from loop corrections to theKahler potentials

63 Instanton corrections to the gauge kinetic functions

So far we have discussed space-time instanton corrections to the superpotentialThese involved one-loop determinants which are given by annulus vacuum di-agrams with at least one E2 boundary These are related to one-loop gaugethreshold corrections to the gauge theory on a D6-brane wrapping the same cycleas the E2 instanton

Now we can ask what other corrections these space-time instantons can induceBy applying S- and T-dualities to the story of world-sheet instanton correctionsin the heterotic string we expect that there can also be E2-instanton corrections

to the holomorphic gauge kinetic functions In the heterotic case similar to thetopological Type II string such corrections arise from string world-sheets of Eulercharacteristic zero ie here from world-sheets with two boundaries Thereforewe expect such corrections to appear for E2-instantons admitting one complexopen string modulus ie those wrapping a three-cycle with Betti number b1(Ξ) =1

Let us start by discussing the instanton zero mode structure for such a cycleFirst let us provide the form of the vertex operators The bosonic fields in the(minus1) ghost picture are

V (minus1)y (z) = y eminusϕ(z)Σh= 1

2q=plusmn1(z) (614)

which before the orientifold projection are accompanied by the two pairs offermionic zero modes

V (minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 1

2(z) (615)

and

V(minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=+ 1

2(z) (616)

Now one has to distinguish two cases depending on how the anti-holomorphicinvolution σ acts on the open string modulus Y

σ y rarr plusmny (617)

In the case that y is invariant under σ called first kind in the following theorientifold projection acts in the same way as for the 4D fields Xmicro ie thetwo bosonic zero modes y and the two fermionic zero modes micro survive In the

20

other case dubbed second kind the bosonic zero mode is projected out andonly the fermionic modulino zero mode micro survives4 Therefore in the absence ofany additional zero modes for instance from E2-D6 intersections the zero modemeasure in any instanton amplitude assumes the following form

intd4x d2θ d2y d2micro eminusSE2 for σ y rarr y (618)

andintd4x d2θ d2micro eminusSE2 for σ y rarr minusy (619)

As an example consider the set-up in figure 3 with σ yi rarr minusyi Here the

x x x2 31

1 2 3y y y

O6

E2

∆x1 ∆ y3

∆x2

Figure 3 Deformations of an instanton which is invariant under the orientifold pro-jection

deformations ∆x12 are of the first kind and ∆y3 is of the second kindNow it is clear that an instanton with precisely one set of fermionic zero

modes of the second kind and no additional zero modes can generate a correctionto the SU(Na) gauge kinetic function The instanton amplitude takes on thefollowing form

〈Fa(p1)Fa(p2)〉E2 =

intd4x d2θ d2micro exp(minusSE2) exp (Z prime

0(E2)) AF 2a(E2 D6a)

where AF 2a(E2 D6a) is the annulus diagram in figure 4 which absorbs all the

appearing fermionic zero modes and where the gauge boson vertex operators inthe (0)-ghost picture have the usual form

V(0)A (z) = ǫmicro (partmicroX(z) + i(p middot ψ)ψmicro(z)) e

ipmiddotX(z) (620)

4By duality this distinction is related to the two kinds of deformations of genus g curvesstudied in [42] The first kind are the curves moving in families ie transversal deformationof the curve The second kind is related to the deformations coming with the genus g of thecurve

21

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

On the right hand side of eq (51) fa denotes the Wilsonian ie holomorphicgauge kinetic function which is given in terms of a holomorphic tree-level functionplus the holomorphic part of the one-loop threshold corrections (cf eq (27))

fa = f treea + f 1minusloop

a

(eminusT c

i

)=sum

I

M IaU

cI +∆hol

a (54)

In addition on the right hand side of eq (51) the non-holomorphic terms propor-tional to the Kahler metric of the moduli K and the matter field Kahler metricsKr are due to the one-loop contributions of massless fields These fields generatenon-local terms in the one-loop effective action which correspond to one-loopnon-invariances under σ-model transformations the so-called σ-model anomalies(Kahler and reparametrisation anomalies)

Matching up all terms in eq (51) essentially means that the σ-model anoma-lies can be cancelled in a two-fold way First by local contributions to the gaugecoupling constant via the one-loop threshold contributions ∆a These terms origi-nate from massive string states The second way to cancel the σ-model anomaliesis due to a field dependent (however gauge group independent) one-loop contribu-tion to the Kahler potential of the chiral moduli fields It implies that some of themoduli fields transform non-trivially under the Kahler transformations and alsounder reparametrisations in moduli space The universal one-loop modificationof the Kahler potential is nothing else than a generalised Green-Schwarz mecha-nism cancelling the σ-model anomalies This is analogous to the Green-Schwarzmechanism which cancels anomalies of physical U(1) gauge fields whereas theσ-model anomalies correspond to unphysical composite gauge connections Ef-fectively it means that the Green-Schwarz mechanism with respect to the σ-modelanomalies can be described by a non-holomorphic one-loop field redefinition ofthe associated tree-level moduli fields

As we will see in type IIA orientifold models these field redefinitions act onthe real parts of the dilaton field S as well as the complex structure moduli UJ

S rarr S + δGS(U T )

UJ rarr UJ + δGSJ (U T ) (55)

These redefined fields are those that determine the gauge coupling constants1g2astring at the string scale Recall that as the tree-level gauge couplings (24)are already gauge group dependent so are these one-loop corrections but theonly dependence arises due to the universal one-loop redefinition of the modulifields It is in this sense that we still call these one-loop corrections to the gaugecouplings universal Note that in heterotic string compactifications the σ-modelGreen-Schwarz mechanism only acts on the heterotic dilaton field

10

51 Holomorphic gauge couplings for toroidal models

In summary equation (51) is to be understood recursively which means that onecan insert the tree-level results into the last three terms of eq (51) In additionone also has to include the universal field redefinition eq (55) in 1g2astring inthe left hand side of eq (51) in order to get a complete matching of all termsin eq (51) as we will demonstrate for the aforementioned toroidal orbifold inthe following For N = 2 sectors the one-loop threshold corrections to the gaugecoupling constant indeed contain a holomorphic Wilsonian term f

(1)a whereas

for N = 1 sectors only the non-holomorphic piece ∆nha is present

Specifically the holomorphic gauge kinetic function can now be determinedby comparing the string theoretical formula (31) for the effective gauge couplingwith the field theoretical one (51) The first thing to notice are the differentcutoff scales appearing in the two formulas One needs to convert one into theother using

M2s

M2Pl

prop exp(2φ4) prop (S U1 U2 U3)minus 1

2 (56)

Here φ4 is the four dimensional dilaton and the complex structure moduli in thesupergravity basis can be expressed in terms of φ4 and the complex structuremoduli uI = RI2RI1 as

S =1

2πeminusφ4

1radicu1 u2 u3

UI =1

2πeminusφ4

radicuJ uKuI

with I 6= J 6= K 6= I (57)

These fields are the real parts of complex scalars of four dimensional chiral mul-tiplets Sc and U c

I As N = 4 super YangndashMills theory is finite one expects the sum of the terms

in (31) proportional to T (Ga) to cancel This is because the only chiral multipletstransforming in the adjoint representation of the gauge group are the open stringmoduli which (on the background considered) assemble themselves into threechiral multiplets thus forming an N = 4 sector together with the gauge fieldsTo show that this cancellation does happen one notices the following Firstlynadjoint = 3 as explained such that there is no term in ba proportional to T (Ga)Secondly

K = minus ln(Sc + Sc)minus

3sum

I=1

ln(U cI + U

c

I)minus3sum

I=1

ln(T cI + T

c

I) (58)

gminus2atree = S

3prod

I=1

nIa minus

3sum

I=1

UI nIam

Jam

Ka I 6= J 6= K 6= I (59)

where TI are the Kahler moduli of the torus and nIa m

Ia are the wrapping numbers

of the brane Finally one needs the matter metric for the open string moduli

11

which can be obtained from the T-dual expression in models with D9- and D5-branes [4] Performing the T-duality which essentially amounts to exchangingKahler and complex structure moduli and converting gauge flux into non-trivialintersection angles for the D6-branes one arrives at (I = 1 3)1

KIij =

δijTIUI

∣∣∣∣∣(nJ

a + iuJ mJa )(n

Ka + iuK m

Ka )

(nIa + iuI mI

a)

∣∣∣∣∣ I 6= J 6= K 6= I (510)

Let us now turn to the fields in the fundamental representation of the gaugegroup Ga in particular to the fields arising from the intersection with one otherstack of branes denoted by b For an N = 1 open string sector the metric forthese fields can be written as [34 4] (see also [35])

Kabij = δij S

minusα

3prod

I=1

Uminus(β+ξ θI

ab)

I Tminus(γ+ζ θI

ab)

I

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (511)

where α β γ ξ and ζ are undetermined constants As θ12ab gt 0 and θ3ab lt 0which is assumed in (511) the intersection number Iab is positive implying that

nf = IabNb (512)

Using Ta(f) =12and relations (52 56 58 511 512) one finds a contribution

to the right hand side of (51) proportional to

IabNb

2

(ln

(M2

s

micro2

)+ (2γ minus 1) ln(T1T2T3) + (2β minus 1

2) ln(U1U2U3) + (2αminus 1

2) lnS

+ζ

3sum

I=1

θIab lnTI + ξ

3sum

I=1

θIab lnUI minus ln

[Γ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab)

]) (513)

Using (36) one finds that the first and the last term exactly reproduce the contri-bution of the last two terms in (31) The terms proportional to ζ and ξ will laterbe shown to constitute the aforementioned universal gauge coupling correctionThe remaining three terms can neither be attributed to such a correction nor canthey be written as the real part of a holomorphic function Thus they cannot bethe one-loop correction to the gauge kinetic function and therefore must vanishThis fixes some of the coefficients in the ansatz (511)

α = β =1

4 γ =

1

2 (514)

1An overall factor involving the wrapping numbers was introduced in this expression in orderto achieve full cancellation This can be done as the expressions used are derived only up tooverall constants [34]

12

The same matching of terms appears between the Mobius diagram plus the annu-lus with boundaries on brane a and its orientifold image and the Kahler metricsfor fields in the symmetric and antisymmetric representation Here one has toreplace θIab and IabNb by θIaaprime = 2θIa and IaaprimeNa in (511) and (33) Apart fromthese replacements the Kahler metric for matter in these representations is alsogiven by (511) with the constants α β γ given in (514)

The corrections to the gauge couplings coming fromN = 2 open string sectorswere seen in the previous section to take on quite a different form They containa term

minus ln |η(i T cI )|4 = minus4Re [ln η(i T c

I )] (515)

which can be written as the real part of a holomorphic function This leads oneto conclude that the gauge kinetic function receives one-loop corrections fromthese sectors Inserting the correct prefactor which from the first term in (35)and the corresponding one in (51) can be seen to be proportional to the betafunction coefficient gives

f (1)a = minusNb |IJab IKab |

4π2ln η(i T c

I ) I 6= J 6= K 6= I (516)

where again I denotes the torus in which the branes lie on top of each other andIJKab are the intersection numbers on the other tori

The term minus ln(TI VaI ) in (35) is not the real part of a holomorphic function

Proceeding as before one finds that the Kahler metric for the hypermultiplet (ortwo chiral multiplets) living at an intersection of branes a and b preserving eightsupercharges must be

KIij =

|nIa + iuI m

Ia|

(UJ UK T J TK)12

I 6= J 6= K 6= I (517)

Apart from the factor in the numerator this is in agreement with the form foundby direct calculations [34 4] The appearance of the numerator is however plau-sible as it also appears in the open string moduli metric and the hypermultipletsunder discussion should feel the Irsquoth torus in the same way

52 Universal threshold corrections

In the following the aforementioned ldquouniversalrdquo gauge coupling corrections willbe discussed They also appear in the heterotic [13] and type I [36 37] string andare related to a redefinition of the dilaton at one-loop [9 8] This stems from thefact that the dilaton really lives in a linear multiplet rather than a chiral one

Our general ansatz for the Kahler metrics for the chiral matter in an N = 1sector contains a factor

3prod

J=1

Uminusξ θJ

ab

J Tminusζ θJ

ab

J (518)

13

which according to (51) appears in the one-loop correction to the gauge couplingconstant Neither is this term reproduced in the string one-loop calculation ofthe coupling nor can it be written as a correction to the holomorphic gaugekinetic function Therefore as is familiar from gauge threshold computationsthere remains the possibility that it can be absorbed into a one-loop correctionto the S and UI chiral superfields In the following we require that such a gaugegroup factor independent universal correction is possible and see how this fixesthe parameters in (518)

The first observation is that in order to get something gauge group indepen-dent the factor (518) actually must have the following form

3prod

J=1

Uminusξprime sign(Iab)θ

Jab

J Tminusζprime sign(Iab)θ

Jab

J (519)

with ξprime and ζ prime independent of the brane For the metrics of fields transformingin the symmetric or antisymmetric representation of the gauge group one hasto replace φab = φa minus φb by φaaprime = 2φa and sign(Iab) by sign(Iaaprime minus IaO6) orsign(Iaaprime + IaO6) respectively Then one computes (K prime denotes the factor (518519) appearing in the full Kahler metric K)

sum

r

Ta(r) ln detKprimer = |Iab|Nb

2ln

[3prod

J=1

Uminusξprime sign(Iab) θ

Jab

J Tminusζprime sign(Iab) θ

Jab

J

](520)

+|Iabprime |Nb

2ln

[3prod

J=1

Uminusξprime sign(Iabprime) θ

Jabprime

J Tminusζprime sign(Iabprime ) θ

Jabprime

J

]

+ Na+22

|IaaprimeminusIaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(IaaprimeminusIaO6) θ

Ja

J Tminus2ζprime sign(IaaprimeminusIaO6) θ

Ja

J

]

+ Naminus22

|Iaaprime+IaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(Iaaprime+IaO6) θ

Ja

J Tminus2ζprime sign(Iaaprime+IaO6) θ

Ja

J

]

After a few steps using |Iab| sign(Iab) = Iab and the tadpole cancellation condi-tion this can be brought to the simple form

sum

r

Ta(r) ln detKprimer = minusn1

an2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

minus3sum

J=1

nJam

Ka m

La

[sum

b

NbmJb n

Kb n

Lb

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

J 6= K 6= L 6= J (521)

Therefore these corrections have precisely the form required for them to be iden-tified with the one-loop correction between the linear superfields appearing in

14

string theory and the chiral superfields used in the supergravity description

SL = S minus 1

8π2

sum

b

Nbm1bm

2bm

3b

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

ULJ = UJ +

1

8π2

sum

b

NbmJb n

Kb n

Lb

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

J 6= K 6= L 6= J (522)

In contrast to eq (59) where the tree-level gauge couplings are determined theone-loop gauge couplings at the string scale have to include the redefined fieldsSL and UL

gminus2astring = SL

3prod

I=1

nIa minus

3sum

I=1

ULI n

Iam

Jam

Ka (523)

In contrast to all (to us) known cases studied in the literature for ξprime 6= 0 thefields which are corrected ie the moduli S and UI also appear in the one-loopredefinition Let us propose an argument why such corrections might be expectedto be absent Due to the anomalous U(1) gauge symmetries the chiral superfieldsS and UI participate in the Green-Schwarz mechanism and therefore transformnon-trivially under U(1) gauge transformations This implies that in order tobe gauge invariant the one-loop corrections in (522) proportional to lnUI mustbe extended in the usual way by UI rarr UI + δaGSVa Computing the resultingFI-terms via the supergravity formula ξa2g

2a = partKpartVa|Va=0 gives besides the

tree-level result depending on SL ULI a one-loop contribution proportional to

ξprimesum

b ξ(0)b φJ

b UJ This has an extra dependence on the complex structure moduliUI However for intersecting D6-branes we have seen that the Wilsonian (super-gravity) FI-Terms are proportional to the Wilsonian gauge threshold correctionswhich depend only on the Kahler moduli via instanton corrections (for N = 1sectors they are even vanishing in the setup at hand) This seems to suggest thatthere should better be no UminusθI

I dependence in the matter field Kahler metricsie ξprime = 0

Moreover in analogy to the heterotic string we expect that for the rangeminus1 le θJ le 1 the exponent of TJ runs over the range [minus1 0] This conditionwould fix ζ prime = plusmn12

Let us summarise the conclusions we have drawn from requiring holomorphyof the Wilsonian gauge kinetic function First we provided arguments that theKahler metric for N = 1 chiral matter fields in intersecting D6-brane models isof the following form

Kabij = δij S

minus 14

3prod

J=1

Uminus 1

4J T

minus( 12plusmn 1

2sign(Iab) θ

Jab)

J

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (524)

15

where supersymmetry of course requiressum3

I=1 θIab = 0 Second the holomorphic

gauge kinetic function (on the background considered) only receives correctionsfrom N = 2 open string sectors and the one-loop correction takes on the followingform

f (1)a = minus

sum

b

Nb |IJabIKab |4π2

ln η(i T cI ) I 6= J 6= K 6= I (525)

where the sum only runs over branes b which lie on top of brane a in exactly onetorus denoted by I Therefore the results for the gauge threshold corrections andthe matter field Kahler metrics are consistent both with the non-renormalisationtheorem from section 2 and the Kaplunovsky-Louis formula (51) Clearly itwould be interesting along the lines of [34] to carry out a string amplitude com-putation to fix the free coefficient in the ansatz (511) and see whether our indirectarguments are correct

6 Holomorphic E2-instanton amplitudes

Space-time instantons are given also by D-branes which in this case are EuclideanD2-branes (so-called E2-branes) wrapping three-cycles Ξ in the CalabindashYau sothat they are point-like in four-dimensional Minkowski space Such instantonscan contribute to the holomorphic superpotential and gauge kinetic functions onlyif they preserve half of the N = 1 supersymmetry This means that the instantonmeasure must contain a factor d4x d2θ Let us first clarify an important aspectof this half-BPS condition In the second part of this section we then revisitthe computation of contributions of such instantons to the superpotential andalso clarify some issues concerned with the appearing one-loop determinants Inthe third and fourth part we investigate under which conditions such stringinstantons can also contribute to the gauge kinetic functions and FI-terms

61 Half-BPS instantons

As has been explained in [24 18 20] just wrapping an E2-instanton around arigid sLag three-cycle in the Calabi-Yau gives four bosonic and four fermioniczero modes The vertex operators for the latter are

V(minus12)θ (z) = θα e

minusϕ(z)2 (z)Sα(z) Σh= 3

8q= 3

2(z) (61)

and

V(minus12)

θ(z) = θα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 3

2(z) (62)

Therefore if the instanton is not invariant under the orientifold projection onestill has four instead of the desired two fermionic zero modes Thus only by

16

placing the E2-brane in a position invariant under Ωσ does one have a chanceto get rid of the two additional zero modes θ For so called O(n) instantons onecan see that the zero modes xmicro θ are symmetrised and the mode θ gets anti-symmetrised For the opposite projection ie for USp(2n) instantons the zeromodes xmicro θ are anti-symmetrised and the mode θ gets symmetrised Thereforeone can only get the simple d4x d2θ instanton measure for a single O(1) instanton

62 Superpotential contributions

In order to contribute to the superpotential we also require that there do notarise any further zero modes from E2-E2 open strings so that the three-cycleΞ should be rigid ie b1(Ξ) = 0 Therefore considering an E2-instanton in anintersecting brane configuration additional zero modes can only arise from theintersection of the instanton Ξ with D6-branes Πa There are Na [ΞcapΠa]

+ chiralfermionic zero modes λaI and Na [Ξ cap Πa]

minus anti-chiral ones λaJ 2

For its presentation it is useful to introduce the short-hand notation

Φak bk [~xk] = Φak xk1middot Φxk1xk2

middot Φxk2xk3middot middot Φxknminus1xkn

middot Φxkn(k)bk (63)

for the chain-product of open string vertex operators Here we define Φak bk [~0] =Φak bk

To extract the superpotential one can probe it by evaluating an appropriatematter field correlator in the instanton background The CFT allows one tocompute it in physical normalisation which combines the superpotential part Ywith the matter field Kahler metrics like

〈Φa1b1 middot middot ΦaM bM 〉E2minusinst =e

K

2 YΦa1b1ΦaMbMradic

Ka1b1 middot middotKaM bM

(64)

In [14] a general expression for the single E2-instanton contribution to thecharged matter superpotential was proposed involving the evaluation of the fol-lowing zero mode integral over disc and one-loop open string CFT amplitudes

〈Φa1b1 middot middot ΦaM bM 〉E2 =V3gs

intd4x d2θ

sum

conf

proda

(prod[ΞcapΠa]+

i=1 dλia) (prod[ΞcapΠa]minus

i=1 dλi

a

)

exp(minusSE2) exp (Z prime0(E2)) 〈Φa1b1[~x1]〉λa1 λb1

middot middot 〈ΦaLbL[~xL]〉λaLλbL

(65)

For simplicity we do not consider the case that matter fields are also assignedto string loop diagrams The one-loop contributions are annulus diagrams for

2Here we introduced the physical intersection number between two branes Πa cap Πb whichis the sum of positive [Πa capΠb]

+ and negative [Πa cap Πb]minus intersections

17

open strings with one boundary on the E2-instanton and the other boundary onthe various D6-branes and Mobius diagrams with boundary on the E2-instanton

〈1〉1-loop = Z prime0(E2) =

sumb Z

primeA(E2aD6b) + Z primeM(E2aO6) (66)

Here Z prime means that we only sum over the massive open string states in theloop amplitude as the zero modes are taken care of explicitly It was shownthat these instantonic open string loop diagrams are identical to the one-loopthreshold corrections TA(D6aD6b) Diagrammatically we have the intriguingrelation shown in figure 1 and in figure 2 which holds for the even spin structures3

Fa

Fa x

x

D6aaE2 =D6b D6b

Figure 1 Relation between instantonic one-loop amplitudes and corresponding gaugethreshold corrections

Fa

Fa x

x

D6aaE2 =

O6O6

Figure 2 Relation between instantonic Mobius amplitude and corresponding gaugethreshold corrections

The annulus threshold corrections can be computed leading to

ZA(E2aD6b) =

int infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it)

η3(it)ACY

ab [αβ](it) (67)

and the Mobius strip amplitude for the instanton which as we explained mustbe invariant under the orientifold projection yields

ZM(E2aO6) = plusmnint infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it + 1

2

)

η3(it + 1

2

) ACYaa [α

β](it + 1

2

)(68)

3The contribution of the CP-odd Rminus sector is expected to yield corrections to the θ-angle

18

The overall plus sign is for O(1) instantons reflecting the fact that only for thesethe xmicro and θα zero modes survive the orientifold projection Note that up to theargument the Mobius thresholds are ZA(E2aD6a) Therefore for rigid branesthe massless sector reflects the number of four bosonic and two fermionic zeromodes In section 63 we will discuss the number of zero modes if b1(Ξ) gt 0 Allthese stringy threshold corrections are known to be non-holomorphic Thereforeit is not immediately obvious in which sense the expression (65) is meant andhow one can extract the holomorphic superpotential part Y from it

The CFT disc amplitudes in (65) are also not holomorphic but combinenon-holomorphic Kahler potential contributions and holomorphic superpotentialcontributions in the usual way [38 39 40 41]

〈Φab[~x]〉λaλb=

eK

2 YλaΦax1Φx1x2 ΦxN b λbradicKλaaKax1 KxnbKbλb

(69)

=e

K

2 YλabΦab[x]λbradic

Kλaa Kab[x]Kbλb

(610)

Due to the Kaplunovsky-Louis formula (51) the stringy one-loop amplitudes areknown to include the holomorphic Wilsonian part and contributions from wave-function normalisation Applied to the instanton one-loop amplitudes appearingin Z0(E2a) we write

Z0(E2a) = minus8π2 real(f (1)a )minus ba

2ln

(M2

p

micro2

)minus ca

2Ktree (611)

minus ln

(V3gs

)

tree

+sum

b

|IabNb|2

ln[detKab

]tree

where for the brane and instanton configuration in question the coefficients are

ba =sum

b

|IabNb|2

minus 3 ca =sum

b

|IabNb|2

minus 1 (612)

The constant contributions arise from the Mobius amplitude Inserting (69) and(611) in (65) one realises that the Kahler metrics involving an instanton zeromode and a matter field precisely cancel out so that only the matter metricssurvive as required by the general form (64) Moreover the term exp(K2)comes out just right due to the rule that each disc contains precisely two instantonzero modes The holomorphic piece in (64) can therefore be expressed entirelyin terms of other holomorphic quantities like holomorphic Yukawa couplingsthe holomorphic instanton action and the one-loop holomorphic Wilsonian gaugekinetic function on the E2-brane

YΦa1b1ΦaMbM

=sum

conf

signconf exp(minusSE2)tree exp(minusf (1)

a

)

Yλa1bΦa1b1

[~x1] λb1middot middot Yλa1

bΦaLbL[~xL]λbL

(613)

19

This explicitly shows that knowing the tree-level Kahler potentials computingthe matter field correlator in the instanton background up to one-loop level ings is sufficient to deduce the Wilsonian holomorphic instanton generated super-potential Higher order corrections in gs only come from loop corrections to theKahler potentials

63 Instanton corrections to the gauge kinetic functions

So far we have discussed space-time instanton corrections to the superpotentialThese involved one-loop determinants which are given by annulus vacuum di-agrams with at least one E2 boundary These are related to one-loop gaugethreshold corrections to the gauge theory on a D6-brane wrapping the same cycleas the E2 instanton

Now we can ask what other corrections these space-time instantons can induceBy applying S- and T-dualities to the story of world-sheet instanton correctionsin the heterotic string we expect that there can also be E2-instanton corrections

to the holomorphic gauge kinetic functions In the heterotic case similar to thetopological Type II string such corrections arise from string world-sheets of Eulercharacteristic zero ie here from world-sheets with two boundaries Thereforewe expect such corrections to appear for E2-instantons admitting one complexopen string modulus ie those wrapping a three-cycle with Betti number b1(Ξ) =1

Let us start by discussing the instanton zero mode structure for such a cycleFirst let us provide the form of the vertex operators The bosonic fields in the(minus1) ghost picture are

V (minus1)y (z) = y eminusϕ(z)Σh= 1

2q=plusmn1(z) (614)

which before the orientifold projection are accompanied by the two pairs offermionic zero modes

V (minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 1

2(z) (615)

and

V(minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=+ 1

2(z) (616)

Now one has to distinguish two cases depending on how the anti-holomorphicinvolution σ acts on the open string modulus Y

σ y rarr plusmny (617)

In the case that y is invariant under σ called first kind in the following theorientifold projection acts in the same way as for the 4D fields Xmicro ie thetwo bosonic zero modes y and the two fermionic zero modes micro survive In the

20

other case dubbed second kind the bosonic zero mode is projected out andonly the fermionic modulino zero mode micro survives4 Therefore in the absence ofany additional zero modes for instance from E2-D6 intersections the zero modemeasure in any instanton amplitude assumes the following form

intd4x d2θ d2y d2micro eminusSE2 for σ y rarr y (618)

andintd4x d2θ d2micro eminusSE2 for σ y rarr minusy (619)

As an example consider the set-up in figure 3 with σ yi rarr minusyi Here the

x x x2 31

1 2 3y y y

O6

E2

∆x1 ∆ y3

∆x2

Figure 3 Deformations of an instanton which is invariant under the orientifold pro-jection

deformations ∆x12 are of the first kind and ∆y3 is of the second kindNow it is clear that an instanton with precisely one set of fermionic zero

modes of the second kind and no additional zero modes can generate a correctionto the SU(Na) gauge kinetic function The instanton amplitude takes on thefollowing form

〈Fa(p1)Fa(p2)〉E2 =

intd4x d2θ d2micro exp(minusSE2) exp (Z prime

0(E2)) AF 2a(E2 D6a)

where AF 2a(E2 D6a) is the annulus diagram in figure 4 which absorbs all the

appearing fermionic zero modes and where the gauge boson vertex operators inthe (0)-ghost picture have the usual form

V(0)A (z) = ǫmicro (partmicroX(z) + i(p middot ψ)ψmicro(z)) e

ipmiddotX(z) (620)

4By duality this distinction is related to the two kinds of deformations of genus g curvesstudied in [42] The first kind are the curves moving in families ie transversal deformationof the curve The second kind is related to the deformations coming with the genus g of thecurve

21

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

51 Holomorphic gauge couplings for toroidal models

In summary equation (51) is to be understood recursively which means that onecan insert the tree-level results into the last three terms of eq (51) In additionone also has to include the universal field redefinition eq (55) in 1g2astring inthe left hand side of eq (51) in order to get a complete matching of all termsin eq (51) as we will demonstrate for the aforementioned toroidal orbifold inthe following For N = 2 sectors the one-loop threshold corrections to the gaugecoupling constant indeed contain a holomorphic Wilsonian term f

(1)a whereas

for N = 1 sectors only the non-holomorphic piece ∆nha is present

Specifically the holomorphic gauge kinetic function can now be determinedby comparing the string theoretical formula (31) for the effective gauge couplingwith the field theoretical one (51) The first thing to notice are the differentcutoff scales appearing in the two formulas One needs to convert one into theother using

M2s

M2Pl

prop exp(2φ4) prop (S U1 U2 U3)minus 1

2 (56)

Here φ4 is the four dimensional dilaton and the complex structure moduli in thesupergravity basis can be expressed in terms of φ4 and the complex structuremoduli uI = RI2RI1 as

S =1

2πeminusφ4

1radicu1 u2 u3

UI =1

2πeminusφ4

radicuJ uKuI

with I 6= J 6= K 6= I (57)

These fields are the real parts of complex scalars of four dimensional chiral mul-tiplets Sc and U c

I As N = 4 super YangndashMills theory is finite one expects the sum of the terms

in (31) proportional to T (Ga) to cancel This is because the only chiral multipletstransforming in the adjoint representation of the gauge group are the open stringmoduli which (on the background considered) assemble themselves into threechiral multiplets thus forming an N = 4 sector together with the gauge fieldsTo show that this cancellation does happen one notices the following Firstlynadjoint = 3 as explained such that there is no term in ba proportional to T (Ga)Secondly

K = minus ln(Sc + Sc)minus

3sum

I=1

ln(U cI + U

c

I)minus3sum

I=1

ln(T cI + T

c

I) (58)

gminus2atree = S

3prod

I=1

nIa minus

3sum

I=1

UI nIam

Jam

Ka I 6= J 6= K 6= I (59)

where TI are the Kahler moduli of the torus and nIa m

Ia are the wrapping numbers

of the brane Finally one needs the matter metric for the open string moduli

11

which can be obtained from the T-dual expression in models with D9- and D5-branes [4] Performing the T-duality which essentially amounts to exchangingKahler and complex structure moduli and converting gauge flux into non-trivialintersection angles for the D6-branes one arrives at (I = 1 3)1

KIij =

δijTIUI

∣∣∣∣∣(nJ

a + iuJ mJa )(n

Ka + iuK m

Ka )

(nIa + iuI mI

a)

∣∣∣∣∣ I 6= J 6= K 6= I (510)

Let us now turn to the fields in the fundamental representation of the gaugegroup Ga in particular to the fields arising from the intersection with one otherstack of branes denoted by b For an N = 1 open string sector the metric forthese fields can be written as [34 4] (see also [35])

Kabij = δij S

minusα

3prod

I=1

Uminus(β+ξ θI

ab)

I Tminus(γ+ζ θI

ab)

I

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (511)

where α β γ ξ and ζ are undetermined constants As θ12ab gt 0 and θ3ab lt 0which is assumed in (511) the intersection number Iab is positive implying that

nf = IabNb (512)

Using Ta(f) =12and relations (52 56 58 511 512) one finds a contribution

to the right hand side of (51) proportional to

IabNb

2

(ln

(M2

s

micro2

)+ (2γ minus 1) ln(T1T2T3) + (2β minus 1

2) ln(U1U2U3) + (2αminus 1

2) lnS

+ζ

3sum

I=1

θIab lnTI + ξ

3sum

I=1

θIab lnUI minus ln

[Γ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab)

]) (513)

Using (36) one finds that the first and the last term exactly reproduce the contri-bution of the last two terms in (31) The terms proportional to ζ and ξ will laterbe shown to constitute the aforementioned universal gauge coupling correctionThe remaining three terms can neither be attributed to such a correction nor canthey be written as the real part of a holomorphic function Thus they cannot bethe one-loop correction to the gauge kinetic function and therefore must vanishThis fixes some of the coefficients in the ansatz (511)

α = β =1

4 γ =

1

2 (514)

1An overall factor involving the wrapping numbers was introduced in this expression in orderto achieve full cancellation This can be done as the expressions used are derived only up tooverall constants [34]

12

The same matching of terms appears between the Mobius diagram plus the annu-lus with boundaries on brane a and its orientifold image and the Kahler metricsfor fields in the symmetric and antisymmetric representation Here one has toreplace θIab and IabNb by θIaaprime = 2θIa and IaaprimeNa in (511) and (33) Apart fromthese replacements the Kahler metric for matter in these representations is alsogiven by (511) with the constants α β γ given in (514)

The corrections to the gauge couplings coming fromN = 2 open string sectorswere seen in the previous section to take on quite a different form They containa term

minus ln |η(i T cI )|4 = minus4Re [ln η(i T c

I )] (515)

which can be written as the real part of a holomorphic function This leads oneto conclude that the gauge kinetic function receives one-loop corrections fromthese sectors Inserting the correct prefactor which from the first term in (35)and the corresponding one in (51) can be seen to be proportional to the betafunction coefficient gives

f (1)a = minusNb |IJab IKab |

4π2ln η(i T c

I ) I 6= J 6= K 6= I (516)

where again I denotes the torus in which the branes lie on top of each other andIJKab are the intersection numbers on the other tori

The term minus ln(TI VaI ) in (35) is not the real part of a holomorphic function

Proceeding as before one finds that the Kahler metric for the hypermultiplet (ortwo chiral multiplets) living at an intersection of branes a and b preserving eightsupercharges must be

KIij =

|nIa + iuI m

Ia|

(UJ UK T J TK)12

I 6= J 6= K 6= I (517)

Apart from the factor in the numerator this is in agreement with the form foundby direct calculations [34 4] The appearance of the numerator is however plau-sible as it also appears in the open string moduli metric and the hypermultipletsunder discussion should feel the Irsquoth torus in the same way

52 Universal threshold corrections

In the following the aforementioned ldquouniversalrdquo gauge coupling corrections willbe discussed They also appear in the heterotic [13] and type I [36 37] string andare related to a redefinition of the dilaton at one-loop [9 8] This stems from thefact that the dilaton really lives in a linear multiplet rather than a chiral one

Our general ansatz for the Kahler metrics for the chiral matter in an N = 1sector contains a factor

3prod

J=1

Uminusξ θJ

ab

J Tminusζ θJ

ab

J (518)

13

which according to (51) appears in the one-loop correction to the gauge couplingconstant Neither is this term reproduced in the string one-loop calculation ofthe coupling nor can it be written as a correction to the holomorphic gaugekinetic function Therefore as is familiar from gauge threshold computationsthere remains the possibility that it can be absorbed into a one-loop correctionto the S and UI chiral superfields In the following we require that such a gaugegroup factor independent universal correction is possible and see how this fixesthe parameters in (518)

The first observation is that in order to get something gauge group indepen-dent the factor (518) actually must have the following form

3prod

J=1

Uminusξprime sign(Iab)θ

Jab

J Tminusζprime sign(Iab)θ

Jab

J (519)

with ξprime and ζ prime independent of the brane For the metrics of fields transformingin the symmetric or antisymmetric representation of the gauge group one hasto replace φab = φa minus φb by φaaprime = 2φa and sign(Iab) by sign(Iaaprime minus IaO6) orsign(Iaaprime + IaO6) respectively Then one computes (K prime denotes the factor (518519) appearing in the full Kahler metric K)

sum

r

Ta(r) ln detKprimer = |Iab|Nb

2ln

[3prod

J=1

Uminusξprime sign(Iab) θ

Jab

J Tminusζprime sign(Iab) θ

Jab

J

](520)

+|Iabprime |Nb

2ln

[3prod

J=1

Uminusξprime sign(Iabprime) θ

Jabprime

J Tminusζprime sign(Iabprime ) θ

Jabprime

J

]

+ Na+22

|IaaprimeminusIaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(IaaprimeminusIaO6) θ

Ja

J Tminus2ζprime sign(IaaprimeminusIaO6) θ

Ja

J

]

+ Naminus22

|Iaaprime+IaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(Iaaprime+IaO6) θ

Ja

J Tminus2ζprime sign(Iaaprime+IaO6) θ

Ja

J

]

After a few steps using |Iab| sign(Iab) = Iab and the tadpole cancellation condi-tion this can be brought to the simple form

sum

r

Ta(r) ln detKprimer = minusn1

an2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

minus3sum

J=1

nJam

Ka m

La

[sum

b

NbmJb n

Kb n

Lb

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

J 6= K 6= L 6= J (521)

Therefore these corrections have precisely the form required for them to be iden-tified with the one-loop correction between the linear superfields appearing in

14

string theory and the chiral superfields used in the supergravity description

SL = S minus 1

8π2

sum

b

Nbm1bm

2bm

3b

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

ULJ = UJ +

1

8π2

sum

b

NbmJb n

Kb n

Lb

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

J 6= K 6= L 6= J (522)

In contrast to eq (59) where the tree-level gauge couplings are determined theone-loop gauge couplings at the string scale have to include the redefined fieldsSL and UL

gminus2astring = SL

3prod

I=1

nIa minus

3sum

I=1

ULI n

Iam

Jam

Ka (523)

In contrast to all (to us) known cases studied in the literature for ξprime 6= 0 thefields which are corrected ie the moduli S and UI also appear in the one-loopredefinition Let us propose an argument why such corrections might be expectedto be absent Due to the anomalous U(1) gauge symmetries the chiral superfieldsS and UI participate in the Green-Schwarz mechanism and therefore transformnon-trivially under U(1) gauge transformations This implies that in order tobe gauge invariant the one-loop corrections in (522) proportional to lnUI mustbe extended in the usual way by UI rarr UI + δaGSVa Computing the resultingFI-terms via the supergravity formula ξa2g

2a = partKpartVa|Va=0 gives besides the

tree-level result depending on SL ULI a one-loop contribution proportional to

ξprimesum

b ξ(0)b φJ

b UJ This has an extra dependence on the complex structure moduliUI However for intersecting D6-branes we have seen that the Wilsonian (super-gravity) FI-Terms are proportional to the Wilsonian gauge threshold correctionswhich depend only on the Kahler moduli via instanton corrections (for N = 1sectors they are even vanishing in the setup at hand) This seems to suggest thatthere should better be no UminusθI

I dependence in the matter field Kahler metricsie ξprime = 0

Moreover in analogy to the heterotic string we expect that for the rangeminus1 le θJ le 1 the exponent of TJ runs over the range [minus1 0] This conditionwould fix ζ prime = plusmn12

Let us summarise the conclusions we have drawn from requiring holomorphyof the Wilsonian gauge kinetic function First we provided arguments that theKahler metric for N = 1 chiral matter fields in intersecting D6-brane models isof the following form

Kabij = δij S

minus 14

3prod

J=1

Uminus 1

4J T

minus( 12plusmn 1

2sign(Iab) θ

Jab)

J

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (524)

15

where supersymmetry of course requiressum3

I=1 θIab = 0 Second the holomorphic

gauge kinetic function (on the background considered) only receives correctionsfrom N = 2 open string sectors and the one-loop correction takes on the followingform

f (1)a = minus

sum

b

Nb |IJabIKab |4π2

ln η(i T cI ) I 6= J 6= K 6= I (525)

where the sum only runs over branes b which lie on top of brane a in exactly onetorus denoted by I Therefore the results for the gauge threshold corrections andthe matter field Kahler metrics are consistent both with the non-renormalisationtheorem from section 2 and the Kaplunovsky-Louis formula (51) Clearly itwould be interesting along the lines of [34] to carry out a string amplitude com-putation to fix the free coefficient in the ansatz (511) and see whether our indirectarguments are correct

6 Holomorphic E2-instanton amplitudes

Space-time instantons are given also by D-branes which in this case are EuclideanD2-branes (so-called E2-branes) wrapping three-cycles Ξ in the CalabindashYau sothat they are point-like in four-dimensional Minkowski space Such instantonscan contribute to the holomorphic superpotential and gauge kinetic functions onlyif they preserve half of the N = 1 supersymmetry This means that the instantonmeasure must contain a factor d4x d2θ Let us first clarify an important aspectof this half-BPS condition In the second part of this section we then revisitthe computation of contributions of such instantons to the superpotential andalso clarify some issues concerned with the appearing one-loop determinants Inthe third and fourth part we investigate under which conditions such stringinstantons can also contribute to the gauge kinetic functions and FI-terms

61 Half-BPS instantons

As has been explained in [24 18 20] just wrapping an E2-instanton around arigid sLag three-cycle in the Calabi-Yau gives four bosonic and four fermioniczero modes The vertex operators for the latter are

V(minus12)θ (z) = θα e

minusϕ(z)2 (z)Sα(z) Σh= 3

8q= 3

2(z) (61)

and

V(minus12)

θ(z) = θα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 3

2(z) (62)

Therefore if the instanton is not invariant under the orientifold projection onestill has four instead of the desired two fermionic zero modes Thus only by

16

placing the E2-brane in a position invariant under Ωσ does one have a chanceto get rid of the two additional zero modes θ For so called O(n) instantons onecan see that the zero modes xmicro θ are symmetrised and the mode θ gets anti-symmetrised For the opposite projection ie for USp(2n) instantons the zeromodes xmicro θ are anti-symmetrised and the mode θ gets symmetrised Thereforeone can only get the simple d4x d2θ instanton measure for a single O(1) instanton

62 Superpotential contributions

In order to contribute to the superpotential we also require that there do notarise any further zero modes from E2-E2 open strings so that the three-cycleΞ should be rigid ie b1(Ξ) = 0 Therefore considering an E2-instanton in anintersecting brane configuration additional zero modes can only arise from theintersection of the instanton Ξ with D6-branes Πa There are Na [ΞcapΠa]

+ chiralfermionic zero modes λaI and Na [Ξ cap Πa]

minus anti-chiral ones λaJ 2

For its presentation it is useful to introduce the short-hand notation

Φak bk [~xk] = Φak xk1middot Φxk1xk2

middot Φxk2xk3middot middot Φxknminus1xkn

middot Φxkn(k)bk (63)

for the chain-product of open string vertex operators Here we define Φak bk [~0] =Φak bk

To extract the superpotential one can probe it by evaluating an appropriatematter field correlator in the instanton background The CFT allows one tocompute it in physical normalisation which combines the superpotential part Ywith the matter field Kahler metrics like

〈Φa1b1 middot middot ΦaM bM 〉E2minusinst =e

K

2 YΦa1b1ΦaMbMradic

Ka1b1 middot middotKaM bM

(64)

In [14] a general expression for the single E2-instanton contribution to thecharged matter superpotential was proposed involving the evaluation of the fol-lowing zero mode integral over disc and one-loop open string CFT amplitudes

〈Φa1b1 middot middot ΦaM bM 〉E2 =V3gs

intd4x d2θ

sum

conf

proda

(prod[ΞcapΠa]+

i=1 dλia) (prod[ΞcapΠa]minus

i=1 dλi

a

)

exp(minusSE2) exp (Z prime0(E2)) 〈Φa1b1[~x1]〉λa1 λb1

middot middot 〈ΦaLbL[~xL]〉λaLλbL

(65)

For simplicity we do not consider the case that matter fields are also assignedto string loop diagrams The one-loop contributions are annulus diagrams for

2Here we introduced the physical intersection number between two branes Πa cap Πb whichis the sum of positive [Πa capΠb]

+ and negative [Πa cap Πb]minus intersections

17

open strings with one boundary on the E2-instanton and the other boundary onthe various D6-branes and Mobius diagrams with boundary on the E2-instanton

〈1〉1-loop = Z prime0(E2) =

sumb Z

primeA(E2aD6b) + Z primeM(E2aO6) (66)

Here Z prime means that we only sum over the massive open string states in theloop amplitude as the zero modes are taken care of explicitly It was shownthat these instantonic open string loop diagrams are identical to the one-loopthreshold corrections TA(D6aD6b) Diagrammatically we have the intriguingrelation shown in figure 1 and in figure 2 which holds for the even spin structures3

Fa

Fa x

x

D6aaE2 =D6b D6b

Figure 1 Relation between instantonic one-loop amplitudes and corresponding gaugethreshold corrections

Fa

Fa x

x

D6aaE2 =

O6O6

Figure 2 Relation between instantonic Mobius amplitude and corresponding gaugethreshold corrections

The annulus threshold corrections can be computed leading to

ZA(E2aD6b) =

int infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it)

η3(it)ACY

ab [αβ](it) (67)

and the Mobius strip amplitude for the instanton which as we explained mustbe invariant under the orientifold projection yields

ZM(E2aO6) = plusmnint infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it + 1

2

)

η3(it + 1

2

) ACYaa [α

β](it + 1

2

)(68)

3The contribution of the CP-odd Rminus sector is expected to yield corrections to the θ-angle

18

The overall plus sign is for O(1) instantons reflecting the fact that only for thesethe xmicro and θα zero modes survive the orientifold projection Note that up to theargument the Mobius thresholds are ZA(E2aD6a) Therefore for rigid branesthe massless sector reflects the number of four bosonic and two fermionic zeromodes In section 63 we will discuss the number of zero modes if b1(Ξ) gt 0 Allthese stringy threshold corrections are known to be non-holomorphic Thereforeit is not immediately obvious in which sense the expression (65) is meant andhow one can extract the holomorphic superpotential part Y from it

The CFT disc amplitudes in (65) are also not holomorphic but combinenon-holomorphic Kahler potential contributions and holomorphic superpotentialcontributions in the usual way [38 39 40 41]

〈Φab[~x]〉λaλb=

eK

2 YλaΦax1Φx1x2 ΦxN b λbradicKλaaKax1 KxnbKbλb

(69)

=e

K

2 YλabΦab[x]λbradic

Kλaa Kab[x]Kbλb

(610)

Due to the Kaplunovsky-Louis formula (51) the stringy one-loop amplitudes areknown to include the holomorphic Wilsonian part and contributions from wave-function normalisation Applied to the instanton one-loop amplitudes appearingin Z0(E2a) we write

Z0(E2a) = minus8π2 real(f (1)a )minus ba

2ln

(M2

p

micro2

)minus ca

2Ktree (611)

minus ln

(V3gs

)

tree

+sum

b

|IabNb|2

ln[detKab

]tree

where for the brane and instanton configuration in question the coefficients are

ba =sum

b

|IabNb|2

minus 3 ca =sum

b

|IabNb|2

minus 1 (612)

The constant contributions arise from the Mobius amplitude Inserting (69) and(611) in (65) one realises that the Kahler metrics involving an instanton zeromode and a matter field precisely cancel out so that only the matter metricssurvive as required by the general form (64) Moreover the term exp(K2)comes out just right due to the rule that each disc contains precisely two instantonzero modes The holomorphic piece in (64) can therefore be expressed entirelyin terms of other holomorphic quantities like holomorphic Yukawa couplingsthe holomorphic instanton action and the one-loop holomorphic Wilsonian gaugekinetic function on the E2-brane

YΦa1b1ΦaMbM

=sum

conf

signconf exp(minusSE2)tree exp(minusf (1)

a

)

Yλa1bΦa1b1

[~x1] λb1middot middot Yλa1

bΦaLbL[~xL]λbL

(613)

19

This explicitly shows that knowing the tree-level Kahler potentials computingthe matter field correlator in the instanton background up to one-loop level ings is sufficient to deduce the Wilsonian holomorphic instanton generated super-potential Higher order corrections in gs only come from loop corrections to theKahler potentials

63 Instanton corrections to the gauge kinetic functions

So far we have discussed space-time instanton corrections to the superpotentialThese involved one-loop determinants which are given by annulus vacuum di-agrams with at least one E2 boundary These are related to one-loop gaugethreshold corrections to the gauge theory on a D6-brane wrapping the same cycleas the E2 instanton

Now we can ask what other corrections these space-time instantons can induceBy applying S- and T-dualities to the story of world-sheet instanton correctionsin the heterotic string we expect that there can also be E2-instanton corrections

to the holomorphic gauge kinetic functions In the heterotic case similar to thetopological Type II string such corrections arise from string world-sheets of Eulercharacteristic zero ie here from world-sheets with two boundaries Thereforewe expect such corrections to appear for E2-instantons admitting one complexopen string modulus ie those wrapping a three-cycle with Betti number b1(Ξ) =1

Let us start by discussing the instanton zero mode structure for such a cycleFirst let us provide the form of the vertex operators The bosonic fields in the(minus1) ghost picture are

V (minus1)y (z) = y eminusϕ(z)Σh= 1

2q=plusmn1(z) (614)

which before the orientifold projection are accompanied by the two pairs offermionic zero modes

V (minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 1

2(z) (615)

and

V(minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=+ 1

2(z) (616)

Now one has to distinguish two cases depending on how the anti-holomorphicinvolution σ acts on the open string modulus Y

σ y rarr plusmny (617)

In the case that y is invariant under σ called first kind in the following theorientifold projection acts in the same way as for the 4D fields Xmicro ie thetwo bosonic zero modes y and the two fermionic zero modes micro survive In the

20

other case dubbed second kind the bosonic zero mode is projected out andonly the fermionic modulino zero mode micro survives4 Therefore in the absence ofany additional zero modes for instance from E2-D6 intersections the zero modemeasure in any instanton amplitude assumes the following form

intd4x d2θ d2y d2micro eminusSE2 for σ y rarr y (618)

andintd4x d2θ d2micro eminusSE2 for σ y rarr minusy (619)

As an example consider the set-up in figure 3 with σ yi rarr minusyi Here the

x x x2 31

1 2 3y y y

O6

E2

∆x1 ∆ y3

∆x2

Figure 3 Deformations of an instanton which is invariant under the orientifold pro-jection

deformations ∆x12 are of the first kind and ∆y3 is of the second kindNow it is clear that an instanton with precisely one set of fermionic zero

modes of the second kind and no additional zero modes can generate a correctionto the SU(Na) gauge kinetic function The instanton amplitude takes on thefollowing form

〈Fa(p1)Fa(p2)〉E2 =

intd4x d2θ d2micro exp(minusSE2) exp (Z prime

0(E2)) AF 2a(E2 D6a)

where AF 2a(E2 D6a) is the annulus diagram in figure 4 which absorbs all the

appearing fermionic zero modes and where the gauge boson vertex operators inthe (0)-ghost picture have the usual form

V(0)A (z) = ǫmicro (partmicroX(z) + i(p middot ψ)ψmicro(z)) e

ipmiddotX(z) (620)

4By duality this distinction is related to the two kinds of deformations of genus g curvesstudied in [42] The first kind are the curves moving in families ie transversal deformationof the curve The second kind is related to the deformations coming with the genus g of thecurve

21

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

which can be obtained from the T-dual expression in models with D9- and D5-branes [4] Performing the T-duality which essentially amounts to exchangingKahler and complex structure moduli and converting gauge flux into non-trivialintersection angles for the D6-branes one arrives at (I = 1 3)1

KIij =

δijTIUI

∣∣∣∣∣(nJ

a + iuJ mJa )(n

Ka + iuK m

Ka )

(nIa + iuI mI

a)

∣∣∣∣∣ I 6= J 6= K 6= I (510)

Let us now turn to the fields in the fundamental representation of the gaugegroup Ga in particular to the fields arising from the intersection with one otherstack of branes denoted by b For an N = 1 open string sector the metric forthese fields can be written as [34 4] (see also [35])

Kabij = δij S

minusα

3prod

I=1

Uminus(β+ξ θI

ab)

I Tminus(γ+ζ θI

ab)

I

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (511)

where α β γ ξ and ζ are undetermined constants As θ12ab gt 0 and θ3ab lt 0which is assumed in (511) the intersection number Iab is positive implying that

nf = IabNb (512)

Using Ta(f) =12and relations (52 56 58 511 512) one finds a contribution

to the right hand side of (51) proportional to

IabNb

2

(ln

(M2

s

micro2

)+ (2γ minus 1) ln(T1T2T3) + (2β minus 1

2) ln(U1U2U3) + (2αminus 1

2) lnS

+ζ

3sum

I=1

θIab lnTI + ξ

3sum

I=1

θIab lnUI minus ln

[Γ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab)

]) (513)

Using (36) one finds that the first and the last term exactly reproduce the contri-bution of the last two terms in (31) The terms proportional to ζ and ξ will laterbe shown to constitute the aforementioned universal gauge coupling correctionThe remaining three terms can neither be attributed to such a correction nor canthey be written as the real part of a holomorphic function Thus they cannot bethe one-loop correction to the gauge kinetic function and therefore must vanishThis fixes some of the coefficients in the ansatz (511)

α = β =1

4 γ =

1

2 (514)

1An overall factor involving the wrapping numbers was introduced in this expression in orderto achieve full cancellation This can be done as the expressions used are derived only up tooverall constants [34]

12

The same matching of terms appears between the Mobius diagram plus the annu-lus with boundaries on brane a and its orientifold image and the Kahler metricsfor fields in the symmetric and antisymmetric representation Here one has toreplace θIab and IabNb by θIaaprime = 2θIa and IaaprimeNa in (511) and (33) Apart fromthese replacements the Kahler metric for matter in these representations is alsogiven by (511) with the constants α β γ given in (514)

The corrections to the gauge couplings coming fromN = 2 open string sectorswere seen in the previous section to take on quite a different form They containa term

minus ln |η(i T cI )|4 = minus4Re [ln η(i T c

I )] (515)

which can be written as the real part of a holomorphic function This leads oneto conclude that the gauge kinetic function receives one-loop corrections fromthese sectors Inserting the correct prefactor which from the first term in (35)and the corresponding one in (51) can be seen to be proportional to the betafunction coefficient gives

f (1)a = minusNb |IJab IKab |

4π2ln η(i T c

I ) I 6= J 6= K 6= I (516)

where again I denotes the torus in which the branes lie on top of each other andIJKab are the intersection numbers on the other tori

The term minus ln(TI VaI ) in (35) is not the real part of a holomorphic function

Proceeding as before one finds that the Kahler metric for the hypermultiplet (ortwo chiral multiplets) living at an intersection of branes a and b preserving eightsupercharges must be

KIij =

|nIa + iuI m

Ia|

(UJ UK T J TK)12

I 6= J 6= K 6= I (517)

Apart from the factor in the numerator this is in agreement with the form foundby direct calculations [34 4] The appearance of the numerator is however plau-sible as it also appears in the open string moduli metric and the hypermultipletsunder discussion should feel the Irsquoth torus in the same way

52 Universal threshold corrections

In the following the aforementioned ldquouniversalrdquo gauge coupling corrections willbe discussed They also appear in the heterotic [13] and type I [36 37] string andare related to a redefinition of the dilaton at one-loop [9 8] This stems from thefact that the dilaton really lives in a linear multiplet rather than a chiral one

Our general ansatz for the Kahler metrics for the chiral matter in an N = 1sector contains a factor

3prod

J=1

Uminusξ θJ

ab

J Tminusζ θJ

ab

J (518)

13

which according to (51) appears in the one-loop correction to the gauge couplingconstant Neither is this term reproduced in the string one-loop calculation ofthe coupling nor can it be written as a correction to the holomorphic gaugekinetic function Therefore as is familiar from gauge threshold computationsthere remains the possibility that it can be absorbed into a one-loop correctionto the S and UI chiral superfields In the following we require that such a gaugegroup factor independent universal correction is possible and see how this fixesthe parameters in (518)

The first observation is that in order to get something gauge group indepen-dent the factor (518) actually must have the following form

3prod

J=1

Uminusξprime sign(Iab)θ

Jab

J Tminusζprime sign(Iab)θ

Jab

J (519)

with ξprime and ζ prime independent of the brane For the metrics of fields transformingin the symmetric or antisymmetric representation of the gauge group one hasto replace φab = φa minus φb by φaaprime = 2φa and sign(Iab) by sign(Iaaprime minus IaO6) orsign(Iaaprime + IaO6) respectively Then one computes (K prime denotes the factor (518519) appearing in the full Kahler metric K)

sum

r

Ta(r) ln detKprimer = |Iab|Nb

2ln

[3prod

J=1

Uminusξprime sign(Iab) θ

Jab

J Tminusζprime sign(Iab) θ

Jab

J

](520)

+|Iabprime |Nb

2ln

[3prod

J=1

Uminusξprime sign(Iabprime) θ

Jabprime

J Tminusζprime sign(Iabprime ) θ

Jabprime

J

]

+ Na+22

|IaaprimeminusIaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(IaaprimeminusIaO6) θ

Ja

J Tminus2ζprime sign(IaaprimeminusIaO6) θ

Ja

J

]

+ Naminus22

|Iaaprime+IaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(Iaaprime+IaO6) θ

Ja

J Tminus2ζprime sign(Iaaprime+IaO6) θ

Ja

J

]

After a few steps using |Iab| sign(Iab) = Iab and the tadpole cancellation condi-tion this can be brought to the simple form

sum

r

Ta(r) ln detKprimer = minusn1

an2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

minus3sum

J=1

nJam

Ka m

La

[sum

b

NbmJb n

Kb n

Lb

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

J 6= K 6= L 6= J (521)

Therefore these corrections have precisely the form required for them to be iden-tified with the one-loop correction between the linear superfields appearing in

14

string theory and the chiral superfields used in the supergravity description

SL = S minus 1

8π2

sum

b

Nbm1bm

2bm

3b

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

ULJ = UJ +

1

8π2

sum

b

NbmJb n

Kb n

Lb

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

J 6= K 6= L 6= J (522)

In contrast to eq (59) where the tree-level gauge couplings are determined theone-loop gauge couplings at the string scale have to include the redefined fieldsSL and UL

gminus2astring = SL

3prod

I=1

nIa minus

3sum

I=1

ULI n

Iam

Jam

Ka (523)

In contrast to all (to us) known cases studied in the literature for ξprime 6= 0 thefields which are corrected ie the moduli S and UI also appear in the one-loopredefinition Let us propose an argument why such corrections might be expectedto be absent Due to the anomalous U(1) gauge symmetries the chiral superfieldsS and UI participate in the Green-Schwarz mechanism and therefore transformnon-trivially under U(1) gauge transformations This implies that in order tobe gauge invariant the one-loop corrections in (522) proportional to lnUI mustbe extended in the usual way by UI rarr UI + δaGSVa Computing the resultingFI-terms via the supergravity formula ξa2g

2a = partKpartVa|Va=0 gives besides the

tree-level result depending on SL ULI a one-loop contribution proportional to

ξprimesum

b ξ(0)b φJ

b UJ This has an extra dependence on the complex structure moduliUI However for intersecting D6-branes we have seen that the Wilsonian (super-gravity) FI-Terms are proportional to the Wilsonian gauge threshold correctionswhich depend only on the Kahler moduli via instanton corrections (for N = 1sectors they are even vanishing in the setup at hand) This seems to suggest thatthere should better be no UminusθI

I dependence in the matter field Kahler metricsie ξprime = 0

Moreover in analogy to the heterotic string we expect that for the rangeminus1 le θJ le 1 the exponent of TJ runs over the range [minus1 0] This conditionwould fix ζ prime = plusmn12

Let us summarise the conclusions we have drawn from requiring holomorphyof the Wilsonian gauge kinetic function First we provided arguments that theKahler metric for N = 1 chiral matter fields in intersecting D6-brane models isof the following form

Kabij = δij S

minus 14

3prod

J=1

Uminus 1

4J T

minus( 12plusmn 1

2sign(Iab) θ

Jab)

J

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (524)

15

where supersymmetry of course requiressum3

I=1 θIab = 0 Second the holomorphic

gauge kinetic function (on the background considered) only receives correctionsfrom N = 2 open string sectors and the one-loop correction takes on the followingform

f (1)a = minus

sum

b

Nb |IJabIKab |4π2

ln η(i T cI ) I 6= J 6= K 6= I (525)

where the sum only runs over branes b which lie on top of brane a in exactly onetorus denoted by I Therefore the results for the gauge threshold corrections andthe matter field Kahler metrics are consistent both with the non-renormalisationtheorem from section 2 and the Kaplunovsky-Louis formula (51) Clearly itwould be interesting along the lines of [34] to carry out a string amplitude com-putation to fix the free coefficient in the ansatz (511) and see whether our indirectarguments are correct

6 Holomorphic E2-instanton amplitudes

Space-time instantons are given also by D-branes which in this case are EuclideanD2-branes (so-called E2-branes) wrapping three-cycles Ξ in the CalabindashYau sothat they are point-like in four-dimensional Minkowski space Such instantonscan contribute to the holomorphic superpotential and gauge kinetic functions onlyif they preserve half of the N = 1 supersymmetry This means that the instantonmeasure must contain a factor d4x d2θ Let us first clarify an important aspectof this half-BPS condition In the second part of this section we then revisitthe computation of contributions of such instantons to the superpotential andalso clarify some issues concerned with the appearing one-loop determinants Inthe third and fourth part we investigate under which conditions such stringinstantons can also contribute to the gauge kinetic functions and FI-terms

61 Half-BPS instantons

As has been explained in [24 18 20] just wrapping an E2-instanton around arigid sLag three-cycle in the Calabi-Yau gives four bosonic and four fermioniczero modes The vertex operators for the latter are

V(minus12)θ (z) = θα e

minusϕ(z)2 (z)Sα(z) Σh= 3

8q= 3

2(z) (61)

and

V(minus12)

θ(z) = θα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 3

2(z) (62)

Therefore if the instanton is not invariant under the orientifold projection onestill has four instead of the desired two fermionic zero modes Thus only by

16

placing the E2-brane in a position invariant under Ωσ does one have a chanceto get rid of the two additional zero modes θ For so called O(n) instantons onecan see that the zero modes xmicro θ are symmetrised and the mode θ gets anti-symmetrised For the opposite projection ie for USp(2n) instantons the zeromodes xmicro θ are anti-symmetrised and the mode θ gets symmetrised Thereforeone can only get the simple d4x d2θ instanton measure for a single O(1) instanton

62 Superpotential contributions

In order to contribute to the superpotential we also require that there do notarise any further zero modes from E2-E2 open strings so that the three-cycleΞ should be rigid ie b1(Ξ) = 0 Therefore considering an E2-instanton in anintersecting brane configuration additional zero modes can only arise from theintersection of the instanton Ξ with D6-branes Πa There are Na [ΞcapΠa]

+ chiralfermionic zero modes λaI and Na [Ξ cap Πa]

minus anti-chiral ones λaJ 2

For its presentation it is useful to introduce the short-hand notation

Φak bk [~xk] = Φak xk1middot Φxk1xk2

middot Φxk2xk3middot middot Φxknminus1xkn

middot Φxkn(k)bk (63)

for the chain-product of open string vertex operators Here we define Φak bk [~0] =Φak bk

To extract the superpotential one can probe it by evaluating an appropriatematter field correlator in the instanton background The CFT allows one tocompute it in physical normalisation which combines the superpotential part Ywith the matter field Kahler metrics like

〈Φa1b1 middot middot ΦaM bM 〉E2minusinst =e

K

2 YΦa1b1ΦaMbMradic

Ka1b1 middot middotKaM bM

(64)

In [14] a general expression for the single E2-instanton contribution to thecharged matter superpotential was proposed involving the evaluation of the fol-lowing zero mode integral over disc and one-loop open string CFT amplitudes

〈Φa1b1 middot middot ΦaM bM 〉E2 =V3gs

intd4x d2θ

sum

conf

proda

(prod[ΞcapΠa]+

i=1 dλia) (prod[ΞcapΠa]minus

i=1 dλi

a

)

exp(minusSE2) exp (Z prime0(E2)) 〈Φa1b1[~x1]〉λa1 λb1

middot middot 〈ΦaLbL[~xL]〉λaLλbL

(65)

For simplicity we do not consider the case that matter fields are also assignedto string loop diagrams The one-loop contributions are annulus diagrams for

2Here we introduced the physical intersection number between two branes Πa cap Πb whichis the sum of positive [Πa capΠb]

+ and negative [Πa cap Πb]minus intersections

17

open strings with one boundary on the E2-instanton and the other boundary onthe various D6-branes and Mobius diagrams with boundary on the E2-instanton

〈1〉1-loop = Z prime0(E2) =

sumb Z

primeA(E2aD6b) + Z primeM(E2aO6) (66)

Here Z prime means that we only sum over the massive open string states in theloop amplitude as the zero modes are taken care of explicitly It was shownthat these instantonic open string loop diagrams are identical to the one-loopthreshold corrections TA(D6aD6b) Diagrammatically we have the intriguingrelation shown in figure 1 and in figure 2 which holds for the even spin structures3

Fa

Fa x

x

D6aaE2 =D6b D6b

Figure 1 Relation between instantonic one-loop amplitudes and corresponding gaugethreshold corrections

Fa

Fa x

x

D6aaE2 =

O6O6

Figure 2 Relation between instantonic Mobius amplitude and corresponding gaugethreshold corrections

The annulus threshold corrections can be computed leading to

ZA(E2aD6b) =

int infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it)

η3(it)ACY

ab [αβ](it) (67)

and the Mobius strip amplitude for the instanton which as we explained mustbe invariant under the orientifold projection yields

ZM(E2aO6) = plusmnint infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it + 1

2

)

η3(it + 1

2

) ACYaa [α

β](it + 1

2

)(68)

3The contribution of the CP-odd Rminus sector is expected to yield corrections to the θ-angle

18

The overall plus sign is for O(1) instantons reflecting the fact that only for thesethe xmicro and θα zero modes survive the orientifold projection Note that up to theargument the Mobius thresholds are ZA(E2aD6a) Therefore for rigid branesthe massless sector reflects the number of four bosonic and two fermionic zeromodes In section 63 we will discuss the number of zero modes if b1(Ξ) gt 0 Allthese stringy threshold corrections are known to be non-holomorphic Thereforeit is not immediately obvious in which sense the expression (65) is meant andhow one can extract the holomorphic superpotential part Y from it

The CFT disc amplitudes in (65) are also not holomorphic but combinenon-holomorphic Kahler potential contributions and holomorphic superpotentialcontributions in the usual way [38 39 40 41]

〈Φab[~x]〉λaλb=

eK

2 YλaΦax1Φx1x2 ΦxN b λbradicKλaaKax1 KxnbKbλb

(69)

=e

K

2 YλabΦab[x]λbradic

Kλaa Kab[x]Kbλb

(610)

Due to the Kaplunovsky-Louis formula (51) the stringy one-loop amplitudes areknown to include the holomorphic Wilsonian part and contributions from wave-function normalisation Applied to the instanton one-loop amplitudes appearingin Z0(E2a) we write

Z0(E2a) = minus8π2 real(f (1)a )minus ba

2ln

(M2

p

micro2

)minus ca

2Ktree (611)

minus ln

(V3gs

)

tree

+sum

b

|IabNb|2

ln[detKab

]tree

where for the brane and instanton configuration in question the coefficients are

ba =sum

b

|IabNb|2

minus 3 ca =sum

b

|IabNb|2

minus 1 (612)

The constant contributions arise from the Mobius amplitude Inserting (69) and(611) in (65) one realises that the Kahler metrics involving an instanton zeromode and a matter field precisely cancel out so that only the matter metricssurvive as required by the general form (64) Moreover the term exp(K2)comes out just right due to the rule that each disc contains precisely two instantonzero modes The holomorphic piece in (64) can therefore be expressed entirelyin terms of other holomorphic quantities like holomorphic Yukawa couplingsthe holomorphic instanton action and the one-loop holomorphic Wilsonian gaugekinetic function on the E2-brane

YΦa1b1ΦaMbM

=sum

conf

signconf exp(minusSE2)tree exp(minusf (1)

a

)

Yλa1bΦa1b1

[~x1] λb1middot middot Yλa1

bΦaLbL[~xL]λbL

(613)

19

This explicitly shows that knowing the tree-level Kahler potentials computingthe matter field correlator in the instanton background up to one-loop level ings is sufficient to deduce the Wilsonian holomorphic instanton generated super-potential Higher order corrections in gs only come from loop corrections to theKahler potentials

63 Instanton corrections to the gauge kinetic functions

So far we have discussed space-time instanton corrections to the superpotentialThese involved one-loop determinants which are given by annulus vacuum di-agrams with at least one E2 boundary These are related to one-loop gaugethreshold corrections to the gauge theory on a D6-brane wrapping the same cycleas the E2 instanton

Now we can ask what other corrections these space-time instantons can induceBy applying S- and T-dualities to the story of world-sheet instanton correctionsin the heterotic string we expect that there can also be E2-instanton corrections

to the holomorphic gauge kinetic functions In the heterotic case similar to thetopological Type II string such corrections arise from string world-sheets of Eulercharacteristic zero ie here from world-sheets with two boundaries Thereforewe expect such corrections to appear for E2-instantons admitting one complexopen string modulus ie those wrapping a three-cycle with Betti number b1(Ξ) =1

Let us start by discussing the instanton zero mode structure for such a cycleFirst let us provide the form of the vertex operators The bosonic fields in the(minus1) ghost picture are

V (minus1)y (z) = y eminusϕ(z)Σh= 1

2q=plusmn1(z) (614)

which before the orientifold projection are accompanied by the two pairs offermionic zero modes

V (minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 1

2(z) (615)

and

V(minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=+ 1

2(z) (616)

Now one has to distinguish two cases depending on how the anti-holomorphicinvolution σ acts on the open string modulus Y

σ y rarr plusmny (617)

In the case that y is invariant under σ called first kind in the following theorientifold projection acts in the same way as for the 4D fields Xmicro ie thetwo bosonic zero modes y and the two fermionic zero modes micro survive In the

20

other case dubbed second kind the bosonic zero mode is projected out andonly the fermionic modulino zero mode micro survives4 Therefore in the absence ofany additional zero modes for instance from E2-D6 intersections the zero modemeasure in any instanton amplitude assumes the following form

intd4x d2θ d2y d2micro eminusSE2 for σ y rarr y (618)

andintd4x d2θ d2micro eminusSE2 for σ y rarr minusy (619)

As an example consider the set-up in figure 3 with σ yi rarr minusyi Here the

x x x2 31

1 2 3y y y

O6

E2

∆x1 ∆ y3

∆x2

Figure 3 Deformations of an instanton which is invariant under the orientifold pro-jection

deformations ∆x12 are of the first kind and ∆y3 is of the second kindNow it is clear that an instanton with precisely one set of fermionic zero

modes of the second kind and no additional zero modes can generate a correctionto the SU(Na) gauge kinetic function The instanton amplitude takes on thefollowing form

〈Fa(p1)Fa(p2)〉E2 =

intd4x d2θ d2micro exp(minusSE2) exp (Z prime

0(E2)) AF 2a(E2 D6a)

where AF 2a(E2 D6a) is the annulus diagram in figure 4 which absorbs all the

appearing fermionic zero modes and where the gauge boson vertex operators inthe (0)-ghost picture have the usual form

V(0)A (z) = ǫmicro (partmicroX(z) + i(p middot ψ)ψmicro(z)) e

ipmiddotX(z) (620)

4By duality this distinction is related to the two kinds of deformations of genus g curvesstudied in [42] The first kind are the curves moving in families ie transversal deformationof the curve The second kind is related to the deformations coming with the genus g of thecurve

21

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

The same matching of terms appears between the Mobius diagram plus the annu-lus with boundaries on brane a and its orientifold image and the Kahler metricsfor fields in the symmetric and antisymmetric representation Here one has toreplace θIab and IabNb by θIaaprime = 2θIa and IaaprimeNa in (511) and (33) Apart fromthese replacements the Kahler metric for matter in these representations is alsogiven by (511) with the constants α β γ given in (514)

The corrections to the gauge couplings coming fromN = 2 open string sectorswere seen in the previous section to take on quite a different form They containa term

minus ln |η(i T cI )|4 = minus4Re [ln η(i T c

I )] (515)

which can be written as the real part of a holomorphic function This leads oneto conclude that the gauge kinetic function receives one-loop corrections fromthese sectors Inserting the correct prefactor which from the first term in (35)and the corresponding one in (51) can be seen to be proportional to the betafunction coefficient gives

f (1)a = minusNb |IJab IKab |

4π2ln η(i T c

I ) I 6= J 6= K 6= I (516)

where again I denotes the torus in which the branes lie on top of each other andIJKab are the intersection numbers on the other tori

The term minus ln(TI VaI ) in (35) is not the real part of a holomorphic function

Proceeding as before one finds that the Kahler metric for the hypermultiplet (ortwo chiral multiplets) living at an intersection of branes a and b preserving eightsupercharges must be

KIij =

|nIa + iuI m

Ia|

(UJ UK T J TK)12

I 6= J 6= K 6= I (517)

Apart from the factor in the numerator this is in agreement with the form foundby direct calculations [34 4] The appearance of the numerator is however plau-sible as it also appears in the open string moduli metric and the hypermultipletsunder discussion should feel the Irsquoth torus in the same way

52 Universal threshold corrections

In the following the aforementioned ldquouniversalrdquo gauge coupling corrections willbe discussed They also appear in the heterotic [13] and type I [36 37] string andare related to a redefinition of the dilaton at one-loop [9 8] This stems from thefact that the dilaton really lives in a linear multiplet rather than a chiral one

Our general ansatz for the Kahler metrics for the chiral matter in an N = 1sector contains a factor

3prod

J=1

Uminusξ θJ

ab

J Tminusζ θJ

ab

J (518)

13

which according to (51) appears in the one-loop correction to the gauge couplingconstant Neither is this term reproduced in the string one-loop calculation ofthe coupling nor can it be written as a correction to the holomorphic gaugekinetic function Therefore as is familiar from gauge threshold computationsthere remains the possibility that it can be absorbed into a one-loop correctionto the S and UI chiral superfields In the following we require that such a gaugegroup factor independent universal correction is possible and see how this fixesthe parameters in (518)

The first observation is that in order to get something gauge group indepen-dent the factor (518) actually must have the following form

3prod

J=1

Uminusξprime sign(Iab)θ

Jab

J Tminusζprime sign(Iab)θ

Jab

J (519)

with ξprime and ζ prime independent of the brane For the metrics of fields transformingin the symmetric or antisymmetric representation of the gauge group one hasto replace φab = φa minus φb by φaaprime = 2φa and sign(Iab) by sign(Iaaprime minus IaO6) orsign(Iaaprime + IaO6) respectively Then one computes (K prime denotes the factor (518519) appearing in the full Kahler metric K)

sum

r

Ta(r) ln detKprimer = |Iab|Nb

2ln

[3prod

J=1

Uminusξprime sign(Iab) θ

Jab

J Tminusζprime sign(Iab) θ

Jab

J

](520)

+|Iabprime |Nb

2ln

[3prod

J=1

Uminusξprime sign(Iabprime) θ

Jabprime

J Tminusζprime sign(Iabprime ) θ

Jabprime

J

]

+ Na+22

|IaaprimeminusIaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(IaaprimeminusIaO6) θ

Ja

J Tminus2ζprime sign(IaaprimeminusIaO6) θ

Ja

J

]

+ Naminus22

|Iaaprime+IaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(Iaaprime+IaO6) θ

Ja

J Tminus2ζprime sign(Iaaprime+IaO6) θ

Ja

J

]

After a few steps using |Iab| sign(Iab) = Iab and the tadpole cancellation condi-tion this can be brought to the simple form

sum

r

Ta(r) ln detKprimer = minusn1

an2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

minus3sum

J=1

nJam

Ka m

La

[sum

b

NbmJb n

Kb n

Lb

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

J 6= K 6= L 6= J (521)

Therefore these corrections have precisely the form required for them to be iden-tified with the one-loop correction between the linear superfields appearing in

14

string theory and the chiral superfields used in the supergravity description

SL = S minus 1

8π2

sum

b

Nbm1bm

2bm

3b

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

ULJ = UJ +

1

8π2

sum

b

NbmJb n

Kb n

Lb

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

J 6= K 6= L 6= J (522)

In contrast to eq (59) where the tree-level gauge couplings are determined theone-loop gauge couplings at the string scale have to include the redefined fieldsSL and UL

gminus2astring = SL

3prod

I=1

nIa minus

3sum

I=1

ULI n

Iam

Jam

Ka (523)

In contrast to all (to us) known cases studied in the literature for ξprime 6= 0 thefields which are corrected ie the moduli S and UI also appear in the one-loopredefinition Let us propose an argument why such corrections might be expectedto be absent Due to the anomalous U(1) gauge symmetries the chiral superfieldsS and UI participate in the Green-Schwarz mechanism and therefore transformnon-trivially under U(1) gauge transformations This implies that in order tobe gauge invariant the one-loop corrections in (522) proportional to lnUI mustbe extended in the usual way by UI rarr UI + δaGSVa Computing the resultingFI-terms via the supergravity formula ξa2g

2a = partKpartVa|Va=0 gives besides the

tree-level result depending on SL ULI a one-loop contribution proportional to

ξprimesum

b ξ(0)b φJ

b UJ This has an extra dependence on the complex structure moduliUI However for intersecting D6-branes we have seen that the Wilsonian (super-gravity) FI-Terms are proportional to the Wilsonian gauge threshold correctionswhich depend only on the Kahler moduli via instanton corrections (for N = 1sectors they are even vanishing in the setup at hand) This seems to suggest thatthere should better be no UminusθI

I dependence in the matter field Kahler metricsie ξprime = 0

Moreover in analogy to the heterotic string we expect that for the rangeminus1 le θJ le 1 the exponent of TJ runs over the range [minus1 0] This conditionwould fix ζ prime = plusmn12

Let us summarise the conclusions we have drawn from requiring holomorphyof the Wilsonian gauge kinetic function First we provided arguments that theKahler metric for N = 1 chiral matter fields in intersecting D6-brane models isof the following form

Kabij = δij S

minus 14

3prod

J=1

Uminus 1

4J T

minus( 12plusmn 1

2sign(Iab) θ

Jab)

J

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (524)

15

where supersymmetry of course requiressum3

I=1 θIab = 0 Second the holomorphic

gauge kinetic function (on the background considered) only receives correctionsfrom N = 2 open string sectors and the one-loop correction takes on the followingform

f (1)a = minus

sum

b

Nb |IJabIKab |4π2

ln η(i T cI ) I 6= J 6= K 6= I (525)

where the sum only runs over branes b which lie on top of brane a in exactly onetorus denoted by I Therefore the results for the gauge threshold corrections andthe matter field Kahler metrics are consistent both with the non-renormalisationtheorem from section 2 and the Kaplunovsky-Louis formula (51) Clearly itwould be interesting along the lines of [34] to carry out a string amplitude com-putation to fix the free coefficient in the ansatz (511) and see whether our indirectarguments are correct

6 Holomorphic E2-instanton amplitudes

Space-time instantons are given also by D-branes which in this case are EuclideanD2-branes (so-called E2-branes) wrapping three-cycles Ξ in the CalabindashYau sothat they are point-like in four-dimensional Minkowski space Such instantonscan contribute to the holomorphic superpotential and gauge kinetic functions onlyif they preserve half of the N = 1 supersymmetry This means that the instantonmeasure must contain a factor d4x d2θ Let us first clarify an important aspectof this half-BPS condition In the second part of this section we then revisitthe computation of contributions of such instantons to the superpotential andalso clarify some issues concerned with the appearing one-loop determinants Inthe third and fourth part we investigate under which conditions such stringinstantons can also contribute to the gauge kinetic functions and FI-terms

61 Half-BPS instantons

As has been explained in [24 18 20] just wrapping an E2-instanton around arigid sLag three-cycle in the Calabi-Yau gives four bosonic and four fermioniczero modes The vertex operators for the latter are

V(minus12)θ (z) = θα e

minusϕ(z)2 (z)Sα(z) Σh= 3

8q= 3

2(z) (61)

and

V(minus12)

θ(z) = θα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 3

2(z) (62)

Therefore if the instanton is not invariant under the orientifold projection onestill has four instead of the desired two fermionic zero modes Thus only by

16

placing the E2-brane in a position invariant under Ωσ does one have a chanceto get rid of the two additional zero modes θ For so called O(n) instantons onecan see that the zero modes xmicro θ are symmetrised and the mode θ gets anti-symmetrised For the opposite projection ie for USp(2n) instantons the zeromodes xmicro θ are anti-symmetrised and the mode θ gets symmetrised Thereforeone can only get the simple d4x d2θ instanton measure for a single O(1) instanton

62 Superpotential contributions

In order to contribute to the superpotential we also require that there do notarise any further zero modes from E2-E2 open strings so that the three-cycleΞ should be rigid ie b1(Ξ) = 0 Therefore considering an E2-instanton in anintersecting brane configuration additional zero modes can only arise from theintersection of the instanton Ξ with D6-branes Πa There are Na [ΞcapΠa]

+ chiralfermionic zero modes λaI and Na [Ξ cap Πa]

minus anti-chiral ones λaJ 2

For its presentation it is useful to introduce the short-hand notation

Φak bk [~xk] = Φak xk1middot Φxk1xk2

middot Φxk2xk3middot middot Φxknminus1xkn

middot Φxkn(k)bk (63)

for the chain-product of open string vertex operators Here we define Φak bk [~0] =Φak bk

To extract the superpotential one can probe it by evaluating an appropriatematter field correlator in the instanton background The CFT allows one tocompute it in physical normalisation which combines the superpotential part Ywith the matter field Kahler metrics like

〈Φa1b1 middot middot ΦaM bM 〉E2minusinst =e

K

2 YΦa1b1ΦaMbMradic

Ka1b1 middot middotKaM bM

(64)

In [14] a general expression for the single E2-instanton contribution to thecharged matter superpotential was proposed involving the evaluation of the fol-lowing zero mode integral over disc and one-loop open string CFT amplitudes

〈Φa1b1 middot middot ΦaM bM 〉E2 =V3gs

intd4x d2θ

sum

conf

proda

(prod[ΞcapΠa]+

i=1 dλia) (prod[ΞcapΠa]minus

i=1 dλi

a

)

exp(minusSE2) exp (Z prime0(E2)) 〈Φa1b1[~x1]〉λa1 λb1

middot middot 〈ΦaLbL[~xL]〉λaLλbL

(65)

For simplicity we do not consider the case that matter fields are also assignedto string loop diagrams The one-loop contributions are annulus diagrams for

2Here we introduced the physical intersection number between two branes Πa cap Πb whichis the sum of positive [Πa capΠb]

+ and negative [Πa cap Πb]minus intersections

17

open strings with one boundary on the E2-instanton and the other boundary onthe various D6-branes and Mobius diagrams with boundary on the E2-instanton

〈1〉1-loop = Z prime0(E2) =

sumb Z

primeA(E2aD6b) + Z primeM(E2aO6) (66)

Here Z prime means that we only sum over the massive open string states in theloop amplitude as the zero modes are taken care of explicitly It was shownthat these instantonic open string loop diagrams are identical to the one-loopthreshold corrections TA(D6aD6b) Diagrammatically we have the intriguingrelation shown in figure 1 and in figure 2 which holds for the even spin structures3

Fa

Fa x

x

D6aaE2 =D6b D6b

Figure 1 Relation between instantonic one-loop amplitudes and corresponding gaugethreshold corrections

Fa

Fa x

x

D6aaE2 =

O6O6

Figure 2 Relation between instantonic Mobius amplitude and corresponding gaugethreshold corrections

The annulus threshold corrections can be computed leading to

ZA(E2aD6b) =

int infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it)

η3(it)ACY

ab [αβ](it) (67)

and the Mobius strip amplitude for the instanton which as we explained mustbe invariant under the orientifold projection yields

ZM(E2aO6) = plusmnint infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it + 1

2

)

η3(it + 1

2

) ACYaa [α

β](it + 1

2

)(68)

3The contribution of the CP-odd Rminus sector is expected to yield corrections to the θ-angle

18

The overall plus sign is for O(1) instantons reflecting the fact that only for thesethe xmicro and θα zero modes survive the orientifold projection Note that up to theargument the Mobius thresholds are ZA(E2aD6a) Therefore for rigid branesthe massless sector reflects the number of four bosonic and two fermionic zeromodes In section 63 we will discuss the number of zero modes if b1(Ξ) gt 0 Allthese stringy threshold corrections are known to be non-holomorphic Thereforeit is not immediately obvious in which sense the expression (65) is meant andhow one can extract the holomorphic superpotential part Y from it

The CFT disc amplitudes in (65) are also not holomorphic but combinenon-holomorphic Kahler potential contributions and holomorphic superpotentialcontributions in the usual way [38 39 40 41]

〈Φab[~x]〉λaλb=

eK

2 YλaΦax1Φx1x2 ΦxN b λbradicKλaaKax1 KxnbKbλb

(69)

=e

K

2 YλabΦab[x]λbradic

Kλaa Kab[x]Kbλb

(610)

Due to the Kaplunovsky-Louis formula (51) the stringy one-loop amplitudes areknown to include the holomorphic Wilsonian part and contributions from wave-function normalisation Applied to the instanton one-loop amplitudes appearingin Z0(E2a) we write

Z0(E2a) = minus8π2 real(f (1)a )minus ba

2ln

(M2

p

micro2

)minus ca

2Ktree (611)

minus ln

(V3gs

)

tree

+sum

b

|IabNb|2

ln[detKab

]tree

where for the brane and instanton configuration in question the coefficients are

ba =sum

b

|IabNb|2

minus 3 ca =sum

b

|IabNb|2

minus 1 (612)

The constant contributions arise from the Mobius amplitude Inserting (69) and(611) in (65) one realises that the Kahler metrics involving an instanton zeromode and a matter field precisely cancel out so that only the matter metricssurvive as required by the general form (64) Moreover the term exp(K2)comes out just right due to the rule that each disc contains precisely two instantonzero modes The holomorphic piece in (64) can therefore be expressed entirelyin terms of other holomorphic quantities like holomorphic Yukawa couplingsthe holomorphic instanton action and the one-loop holomorphic Wilsonian gaugekinetic function on the E2-brane

YΦa1b1ΦaMbM

=sum

conf

signconf exp(minusSE2)tree exp(minusf (1)

a

)

Yλa1bΦa1b1

[~x1] λb1middot middot Yλa1

bΦaLbL[~xL]λbL

(613)

19

This explicitly shows that knowing the tree-level Kahler potentials computingthe matter field correlator in the instanton background up to one-loop level ings is sufficient to deduce the Wilsonian holomorphic instanton generated super-potential Higher order corrections in gs only come from loop corrections to theKahler potentials

63 Instanton corrections to the gauge kinetic functions

So far we have discussed space-time instanton corrections to the superpotentialThese involved one-loop determinants which are given by annulus vacuum di-agrams with at least one E2 boundary These are related to one-loop gaugethreshold corrections to the gauge theory on a D6-brane wrapping the same cycleas the E2 instanton

Now we can ask what other corrections these space-time instantons can induceBy applying S- and T-dualities to the story of world-sheet instanton correctionsin the heterotic string we expect that there can also be E2-instanton corrections

to the holomorphic gauge kinetic functions In the heterotic case similar to thetopological Type II string such corrections arise from string world-sheets of Eulercharacteristic zero ie here from world-sheets with two boundaries Thereforewe expect such corrections to appear for E2-instantons admitting one complexopen string modulus ie those wrapping a three-cycle with Betti number b1(Ξ) =1

Let us start by discussing the instanton zero mode structure for such a cycleFirst let us provide the form of the vertex operators The bosonic fields in the(minus1) ghost picture are

V (minus1)y (z) = y eminusϕ(z)Σh= 1

2q=plusmn1(z) (614)

which before the orientifold projection are accompanied by the two pairs offermionic zero modes

V (minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 1

2(z) (615)

and

V(minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=+ 1

2(z) (616)

Now one has to distinguish two cases depending on how the anti-holomorphicinvolution σ acts on the open string modulus Y

σ y rarr plusmny (617)

In the case that y is invariant under σ called first kind in the following theorientifold projection acts in the same way as for the 4D fields Xmicro ie thetwo bosonic zero modes y and the two fermionic zero modes micro survive In the

20

other case dubbed second kind the bosonic zero mode is projected out andonly the fermionic modulino zero mode micro survives4 Therefore in the absence ofany additional zero modes for instance from E2-D6 intersections the zero modemeasure in any instanton amplitude assumes the following form

intd4x d2θ d2y d2micro eminusSE2 for σ y rarr y (618)

andintd4x d2θ d2micro eminusSE2 for σ y rarr minusy (619)

As an example consider the set-up in figure 3 with σ yi rarr minusyi Here the

x x x2 31

1 2 3y y y

O6

E2

∆x1 ∆ y3

∆x2

Figure 3 Deformations of an instanton which is invariant under the orientifold pro-jection

deformations ∆x12 are of the first kind and ∆y3 is of the second kindNow it is clear that an instanton with precisely one set of fermionic zero

modes of the second kind and no additional zero modes can generate a correctionto the SU(Na) gauge kinetic function The instanton amplitude takes on thefollowing form

〈Fa(p1)Fa(p2)〉E2 =

intd4x d2θ d2micro exp(minusSE2) exp (Z prime

0(E2)) AF 2a(E2 D6a)

where AF 2a(E2 D6a) is the annulus diagram in figure 4 which absorbs all the

appearing fermionic zero modes and where the gauge boson vertex operators inthe (0)-ghost picture have the usual form

V(0)A (z) = ǫmicro (partmicroX(z) + i(p middot ψ)ψmicro(z)) e

ipmiddotX(z) (620)

4By duality this distinction is related to the two kinds of deformations of genus g curvesstudied in [42] The first kind are the curves moving in families ie transversal deformationof the curve The second kind is related to the deformations coming with the genus g of thecurve

21

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

which according to (51) appears in the one-loop correction to the gauge couplingconstant Neither is this term reproduced in the string one-loop calculation ofthe coupling nor can it be written as a correction to the holomorphic gaugekinetic function Therefore as is familiar from gauge threshold computationsthere remains the possibility that it can be absorbed into a one-loop correctionto the S and UI chiral superfields In the following we require that such a gaugegroup factor independent universal correction is possible and see how this fixesthe parameters in (518)

The first observation is that in order to get something gauge group indepen-dent the factor (518) actually must have the following form

3prod

J=1

Uminusξprime sign(Iab)θ

Jab

J Tminusζprime sign(Iab)θ

Jab

J (519)

with ξprime and ζ prime independent of the brane For the metrics of fields transformingin the symmetric or antisymmetric representation of the gauge group one hasto replace φab = φa minus φb by φaaprime = 2φa and sign(Iab) by sign(Iaaprime minus IaO6) orsign(Iaaprime + IaO6) respectively Then one computes (K prime denotes the factor (518519) appearing in the full Kahler metric K)

sum

r

Ta(r) ln detKprimer = |Iab|Nb

2ln

[3prod

J=1

Uminusξprime sign(Iab) θ

Jab

J Tminusζprime sign(Iab) θ

Jab

J

](520)

+|Iabprime |Nb

2ln

[3prod

J=1

Uminusξprime sign(Iabprime) θ

Jabprime

J Tminusζprime sign(Iabprime ) θ

Jabprime

J

]

+ Na+22

|IaaprimeminusIaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(IaaprimeminusIaO6) θ

Ja

J Tminus2ζprime sign(IaaprimeminusIaO6) θ

Ja

J

]

+ Naminus22

|Iaaprime+IaO6|

2ln

[3prod

J=1

Uminus2ξprime sign(Iaaprime+IaO6) θ

Ja

J Tminus2ζprime sign(Iaaprime+IaO6) θ

Ja

J

]

After a few steps using |Iab| sign(Iab) = Iab and the tadpole cancellation condi-tion this can be brought to the simple form

sum

r

Ta(r) ln detKprimer = minusn1

an2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

minus3sum

J=1

nJam

Ka m

La

[sum

b

NbmJb n

Kb n

Lb

3sum

I=1

θIb (ξprime lnUI + ζ prime lnTI)

]

J 6= K 6= L 6= J (521)

Therefore these corrections have precisely the form required for them to be iden-tified with the one-loop correction between the linear superfields appearing in

14

string theory and the chiral superfields used in the supergravity description

SL = S minus 1

8π2

sum

b

Nbm1bm

2bm

3b

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

ULJ = UJ +

1

8π2

sum

b

NbmJb n

Kb n

Lb

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

J 6= K 6= L 6= J (522)

In contrast to eq (59) where the tree-level gauge couplings are determined theone-loop gauge couplings at the string scale have to include the redefined fieldsSL and UL

gminus2astring = SL

3prod

I=1

nIa minus

3sum

I=1

ULI n

Iam

Jam

Ka (523)

In contrast to all (to us) known cases studied in the literature for ξprime 6= 0 thefields which are corrected ie the moduli S and UI also appear in the one-loopredefinition Let us propose an argument why such corrections might be expectedto be absent Due to the anomalous U(1) gauge symmetries the chiral superfieldsS and UI participate in the Green-Schwarz mechanism and therefore transformnon-trivially under U(1) gauge transformations This implies that in order tobe gauge invariant the one-loop corrections in (522) proportional to lnUI mustbe extended in the usual way by UI rarr UI + δaGSVa Computing the resultingFI-terms via the supergravity formula ξa2g

2a = partKpartVa|Va=0 gives besides the

tree-level result depending on SL ULI a one-loop contribution proportional to

ξprimesum

b ξ(0)b φJ

b UJ This has an extra dependence on the complex structure moduliUI However for intersecting D6-branes we have seen that the Wilsonian (super-gravity) FI-Terms are proportional to the Wilsonian gauge threshold correctionswhich depend only on the Kahler moduli via instanton corrections (for N = 1sectors they are even vanishing in the setup at hand) This seems to suggest thatthere should better be no UminusθI

I dependence in the matter field Kahler metricsie ξprime = 0

Moreover in analogy to the heterotic string we expect that for the rangeminus1 le θJ le 1 the exponent of TJ runs over the range [minus1 0] This conditionwould fix ζ prime = plusmn12

Let us summarise the conclusions we have drawn from requiring holomorphyof the Wilsonian gauge kinetic function First we provided arguments that theKahler metric for N = 1 chiral matter fields in intersecting D6-brane models isof the following form

Kabij = δij S

minus 14

3prod

J=1

Uminus 1

4J T

minus( 12plusmn 1

2sign(Iab) θ

Jab)

J

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (524)

15

where supersymmetry of course requiressum3

I=1 θIab = 0 Second the holomorphic

gauge kinetic function (on the background considered) only receives correctionsfrom N = 2 open string sectors and the one-loop correction takes on the followingform

f (1)a = minus

sum

b

Nb |IJabIKab |4π2

ln η(i T cI ) I 6= J 6= K 6= I (525)

where the sum only runs over branes b which lie on top of brane a in exactly onetorus denoted by I Therefore the results for the gauge threshold corrections andthe matter field Kahler metrics are consistent both with the non-renormalisationtheorem from section 2 and the Kaplunovsky-Louis formula (51) Clearly itwould be interesting along the lines of [34] to carry out a string amplitude com-putation to fix the free coefficient in the ansatz (511) and see whether our indirectarguments are correct

6 Holomorphic E2-instanton amplitudes

Space-time instantons are given also by D-branes which in this case are EuclideanD2-branes (so-called E2-branes) wrapping three-cycles Ξ in the CalabindashYau sothat they are point-like in four-dimensional Minkowski space Such instantonscan contribute to the holomorphic superpotential and gauge kinetic functions onlyif they preserve half of the N = 1 supersymmetry This means that the instantonmeasure must contain a factor d4x d2θ Let us first clarify an important aspectof this half-BPS condition In the second part of this section we then revisitthe computation of contributions of such instantons to the superpotential andalso clarify some issues concerned with the appearing one-loop determinants Inthe third and fourth part we investigate under which conditions such stringinstantons can also contribute to the gauge kinetic functions and FI-terms

61 Half-BPS instantons

As has been explained in [24 18 20] just wrapping an E2-instanton around arigid sLag three-cycle in the Calabi-Yau gives four bosonic and four fermioniczero modes The vertex operators for the latter are

V(minus12)θ (z) = θα e

minusϕ(z)2 (z)Sα(z) Σh= 3

8q= 3

2(z) (61)

and

V(minus12)

θ(z) = θα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 3

2(z) (62)

Therefore if the instanton is not invariant under the orientifold projection onestill has four instead of the desired two fermionic zero modes Thus only by

16

placing the E2-brane in a position invariant under Ωσ does one have a chanceto get rid of the two additional zero modes θ For so called O(n) instantons onecan see that the zero modes xmicro θ are symmetrised and the mode θ gets anti-symmetrised For the opposite projection ie for USp(2n) instantons the zeromodes xmicro θ are anti-symmetrised and the mode θ gets symmetrised Thereforeone can only get the simple d4x d2θ instanton measure for a single O(1) instanton

62 Superpotential contributions

In order to contribute to the superpotential we also require that there do notarise any further zero modes from E2-E2 open strings so that the three-cycleΞ should be rigid ie b1(Ξ) = 0 Therefore considering an E2-instanton in anintersecting brane configuration additional zero modes can only arise from theintersection of the instanton Ξ with D6-branes Πa There are Na [ΞcapΠa]

+ chiralfermionic zero modes λaI and Na [Ξ cap Πa]

minus anti-chiral ones λaJ 2

For its presentation it is useful to introduce the short-hand notation

Φak bk [~xk] = Φak xk1middot Φxk1xk2

middot Φxk2xk3middot middot Φxknminus1xkn

middot Φxkn(k)bk (63)

for the chain-product of open string vertex operators Here we define Φak bk [~0] =Φak bk

To extract the superpotential one can probe it by evaluating an appropriatematter field correlator in the instanton background The CFT allows one tocompute it in physical normalisation which combines the superpotential part Ywith the matter field Kahler metrics like

〈Φa1b1 middot middot ΦaM bM 〉E2minusinst =e

K

2 YΦa1b1ΦaMbMradic

Ka1b1 middot middotKaM bM

(64)

In [14] a general expression for the single E2-instanton contribution to thecharged matter superpotential was proposed involving the evaluation of the fol-lowing zero mode integral over disc and one-loop open string CFT amplitudes

〈Φa1b1 middot middot ΦaM bM 〉E2 =V3gs

intd4x d2θ

sum

conf

proda

(prod[ΞcapΠa]+

i=1 dλia) (prod[ΞcapΠa]minus

i=1 dλi

a

)

exp(minusSE2) exp (Z prime0(E2)) 〈Φa1b1[~x1]〉λa1 λb1

middot middot 〈ΦaLbL[~xL]〉λaLλbL

(65)

For simplicity we do not consider the case that matter fields are also assignedto string loop diagrams The one-loop contributions are annulus diagrams for

2Here we introduced the physical intersection number between two branes Πa cap Πb whichis the sum of positive [Πa capΠb]

+ and negative [Πa cap Πb]minus intersections

17

open strings with one boundary on the E2-instanton and the other boundary onthe various D6-branes and Mobius diagrams with boundary on the E2-instanton

〈1〉1-loop = Z prime0(E2) =

sumb Z

primeA(E2aD6b) + Z primeM(E2aO6) (66)

Here Z prime means that we only sum over the massive open string states in theloop amplitude as the zero modes are taken care of explicitly It was shownthat these instantonic open string loop diagrams are identical to the one-loopthreshold corrections TA(D6aD6b) Diagrammatically we have the intriguingrelation shown in figure 1 and in figure 2 which holds for the even spin structures3

Fa

Fa x

x

D6aaE2 =D6b D6b

Figure 1 Relation between instantonic one-loop amplitudes and corresponding gaugethreshold corrections

Fa

Fa x

x

D6aaE2 =

O6O6

Figure 2 Relation between instantonic Mobius amplitude and corresponding gaugethreshold corrections

The annulus threshold corrections can be computed leading to

ZA(E2aD6b) =

int infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it)

η3(it)ACY

ab [αβ](it) (67)

and the Mobius strip amplitude for the instanton which as we explained mustbe invariant under the orientifold projection yields

ZM(E2aO6) = plusmnint infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it + 1

2

)

η3(it + 1

2

) ACYaa [α

β](it + 1

2

)(68)

3The contribution of the CP-odd Rminus sector is expected to yield corrections to the θ-angle

18

The overall plus sign is for O(1) instantons reflecting the fact that only for thesethe xmicro and θα zero modes survive the orientifold projection Note that up to theargument the Mobius thresholds are ZA(E2aD6a) Therefore for rigid branesthe massless sector reflects the number of four bosonic and two fermionic zeromodes In section 63 we will discuss the number of zero modes if b1(Ξ) gt 0 Allthese stringy threshold corrections are known to be non-holomorphic Thereforeit is not immediately obvious in which sense the expression (65) is meant andhow one can extract the holomorphic superpotential part Y from it

The CFT disc amplitudes in (65) are also not holomorphic but combinenon-holomorphic Kahler potential contributions and holomorphic superpotentialcontributions in the usual way [38 39 40 41]

〈Φab[~x]〉λaλb=

eK

2 YλaΦax1Φx1x2 ΦxN b λbradicKλaaKax1 KxnbKbλb

(69)

=e

K

2 YλabΦab[x]λbradic

Kλaa Kab[x]Kbλb

(610)

Due to the Kaplunovsky-Louis formula (51) the stringy one-loop amplitudes areknown to include the holomorphic Wilsonian part and contributions from wave-function normalisation Applied to the instanton one-loop amplitudes appearingin Z0(E2a) we write

Z0(E2a) = minus8π2 real(f (1)a )minus ba

2ln

(M2

p

micro2

)minus ca

2Ktree (611)

minus ln

(V3gs

)

tree

+sum

b

|IabNb|2

ln[detKab

]tree

where for the brane and instanton configuration in question the coefficients are

ba =sum

b

|IabNb|2

minus 3 ca =sum

b

|IabNb|2

minus 1 (612)

The constant contributions arise from the Mobius amplitude Inserting (69) and(611) in (65) one realises that the Kahler metrics involving an instanton zeromode and a matter field precisely cancel out so that only the matter metricssurvive as required by the general form (64) Moreover the term exp(K2)comes out just right due to the rule that each disc contains precisely two instantonzero modes The holomorphic piece in (64) can therefore be expressed entirelyin terms of other holomorphic quantities like holomorphic Yukawa couplingsthe holomorphic instanton action and the one-loop holomorphic Wilsonian gaugekinetic function on the E2-brane

YΦa1b1ΦaMbM

=sum

conf

signconf exp(minusSE2)tree exp(minusf (1)

a

)

Yλa1bΦa1b1

[~x1] λb1middot middot Yλa1

bΦaLbL[~xL]λbL

(613)

19

This explicitly shows that knowing the tree-level Kahler potentials computingthe matter field correlator in the instanton background up to one-loop level ings is sufficient to deduce the Wilsonian holomorphic instanton generated super-potential Higher order corrections in gs only come from loop corrections to theKahler potentials

63 Instanton corrections to the gauge kinetic functions

So far we have discussed space-time instanton corrections to the superpotentialThese involved one-loop determinants which are given by annulus vacuum di-agrams with at least one E2 boundary These are related to one-loop gaugethreshold corrections to the gauge theory on a D6-brane wrapping the same cycleas the E2 instanton

Now we can ask what other corrections these space-time instantons can induceBy applying S- and T-dualities to the story of world-sheet instanton correctionsin the heterotic string we expect that there can also be E2-instanton corrections

to the holomorphic gauge kinetic functions In the heterotic case similar to thetopological Type II string such corrections arise from string world-sheets of Eulercharacteristic zero ie here from world-sheets with two boundaries Thereforewe expect such corrections to appear for E2-instantons admitting one complexopen string modulus ie those wrapping a three-cycle with Betti number b1(Ξ) =1

Let us start by discussing the instanton zero mode structure for such a cycleFirst let us provide the form of the vertex operators The bosonic fields in the(minus1) ghost picture are

V (minus1)y (z) = y eminusϕ(z)Σh= 1

2q=plusmn1(z) (614)

which before the orientifold projection are accompanied by the two pairs offermionic zero modes

V (minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 1

2(z) (615)

and

V(minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=+ 1

2(z) (616)

Now one has to distinguish two cases depending on how the anti-holomorphicinvolution σ acts on the open string modulus Y

σ y rarr plusmny (617)

In the case that y is invariant under σ called first kind in the following theorientifold projection acts in the same way as for the 4D fields Xmicro ie thetwo bosonic zero modes y and the two fermionic zero modes micro survive In the

20

other case dubbed second kind the bosonic zero mode is projected out andonly the fermionic modulino zero mode micro survives4 Therefore in the absence ofany additional zero modes for instance from E2-D6 intersections the zero modemeasure in any instanton amplitude assumes the following form

intd4x d2θ d2y d2micro eminusSE2 for σ y rarr y (618)

andintd4x d2θ d2micro eminusSE2 for σ y rarr minusy (619)

As an example consider the set-up in figure 3 with σ yi rarr minusyi Here the

x x x2 31

1 2 3y y y

O6

E2

∆x1 ∆ y3

∆x2

Figure 3 Deformations of an instanton which is invariant under the orientifold pro-jection

deformations ∆x12 are of the first kind and ∆y3 is of the second kindNow it is clear that an instanton with precisely one set of fermionic zero

modes of the second kind and no additional zero modes can generate a correctionto the SU(Na) gauge kinetic function The instanton amplitude takes on thefollowing form

〈Fa(p1)Fa(p2)〉E2 =

intd4x d2θ d2micro exp(minusSE2) exp (Z prime

0(E2)) AF 2a(E2 D6a)

where AF 2a(E2 D6a) is the annulus diagram in figure 4 which absorbs all the

appearing fermionic zero modes and where the gauge boson vertex operators inthe (0)-ghost picture have the usual form

V(0)A (z) = ǫmicro (partmicroX(z) + i(p middot ψ)ψmicro(z)) e

ipmiddotX(z) (620)

4By duality this distinction is related to the two kinds of deformations of genus g curvesstudied in [42] The first kind are the curves moving in families ie transversal deformationof the curve The second kind is related to the deformations coming with the genus g of thecurve

21

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

string theory and the chiral superfields used in the supergravity description

SL = S minus 1

8π2

sum

b

Nbm1bm

2bm

3b

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

ULJ = UJ +

1

8π2

sum

b

NbmJb n

Kb n

Lb

3sum

I=1

φIb (ξ

prime lnUI + ζ prime lnTI)

J 6= K 6= L 6= J (522)

In contrast to eq (59) where the tree-level gauge couplings are determined theone-loop gauge couplings at the string scale have to include the redefined fieldsSL and UL

gminus2astring = SL

3prod

I=1

nIa minus

3sum

I=1

ULI n

Iam

Jam

Ka (523)

In contrast to all (to us) known cases studied in the literature for ξprime 6= 0 thefields which are corrected ie the moduli S and UI also appear in the one-loopredefinition Let us propose an argument why such corrections might be expectedto be absent Due to the anomalous U(1) gauge symmetries the chiral superfieldsS and UI participate in the Green-Schwarz mechanism and therefore transformnon-trivially under U(1) gauge transformations This implies that in order tobe gauge invariant the one-loop corrections in (522) proportional to lnUI mustbe extended in the usual way by UI rarr UI + δaGSVa Computing the resultingFI-terms via the supergravity formula ξa2g

2a = partKpartVa|Va=0 gives besides the

tree-level result depending on SL ULI a one-loop contribution proportional to

ξprimesum

b ξ(0)b φJ

b UJ This has an extra dependence on the complex structure moduliUI However for intersecting D6-branes we have seen that the Wilsonian (super-gravity) FI-Terms are proportional to the Wilsonian gauge threshold correctionswhich depend only on the Kahler moduli via instanton corrections (for N = 1sectors they are even vanishing in the setup at hand) This seems to suggest thatthere should better be no UminusθI

I dependence in the matter field Kahler metricsie ξprime = 0

Moreover in analogy to the heterotic string we expect that for the rangeminus1 le θJ le 1 the exponent of TJ runs over the range [minus1 0] This conditionwould fix ζ prime = plusmn12

Let us summarise the conclusions we have drawn from requiring holomorphyof the Wilsonian gauge kinetic function First we provided arguments that theKahler metric for N = 1 chiral matter fields in intersecting D6-brane models isof the following form

Kabij = δij S

minus 14

3prod

J=1

Uminus 1

4J T

minus( 12plusmn 1

2sign(Iab) θ

Jab)

J

radicΓ(θ1ab)Γ(θ

2ab)Γ(1 + θ3ab)

Γ(1minus θ1ab)Γ(1minus θ2ab)Γ(minusθ3ab) (524)

15

where supersymmetry of course requiressum3

I=1 θIab = 0 Second the holomorphic

gauge kinetic function (on the background considered) only receives correctionsfrom N = 2 open string sectors and the one-loop correction takes on the followingform

f (1)a = minus

sum

b

Nb |IJabIKab |4π2

ln η(i T cI ) I 6= J 6= K 6= I (525)

where the sum only runs over branes b which lie on top of brane a in exactly onetorus denoted by I Therefore the results for the gauge threshold corrections andthe matter field Kahler metrics are consistent both with the non-renormalisationtheorem from section 2 and the Kaplunovsky-Louis formula (51) Clearly itwould be interesting along the lines of [34] to carry out a string amplitude com-putation to fix the free coefficient in the ansatz (511) and see whether our indirectarguments are correct

6 Holomorphic E2-instanton amplitudes

Space-time instantons are given also by D-branes which in this case are EuclideanD2-branes (so-called E2-branes) wrapping three-cycles Ξ in the CalabindashYau sothat they are point-like in four-dimensional Minkowski space Such instantonscan contribute to the holomorphic superpotential and gauge kinetic functions onlyif they preserve half of the N = 1 supersymmetry This means that the instantonmeasure must contain a factor d4x d2θ Let us first clarify an important aspectof this half-BPS condition In the second part of this section we then revisitthe computation of contributions of such instantons to the superpotential andalso clarify some issues concerned with the appearing one-loop determinants Inthe third and fourth part we investigate under which conditions such stringinstantons can also contribute to the gauge kinetic functions and FI-terms

61 Half-BPS instantons

As has been explained in [24 18 20] just wrapping an E2-instanton around arigid sLag three-cycle in the Calabi-Yau gives four bosonic and four fermioniczero modes The vertex operators for the latter are

V(minus12)θ (z) = θα e

minusϕ(z)2 (z)Sα(z) Σh= 3

8q= 3

2(z) (61)

and

V(minus12)

θ(z) = θα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 3

2(z) (62)

Therefore if the instanton is not invariant under the orientifold projection onestill has four instead of the desired two fermionic zero modes Thus only by

16

placing the E2-brane in a position invariant under Ωσ does one have a chanceto get rid of the two additional zero modes θ For so called O(n) instantons onecan see that the zero modes xmicro θ are symmetrised and the mode θ gets anti-symmetrised For the opposite projection ie for USp(2n) instantons the zeromodes xmicro θ are anti-symmetrised and the mode θ gets symmetrised Thereforeone can only get the simple d4x d2θ instanton measure for a single O(1) instanton

62 Superpotential contributions

In order to contribute to the superpotential we also require that there do notarise any further zero modes from E2-E2 open strings so that the three-cycleΞ should be rigid ie b1(Ξ) = 0 Therefore considering an E2-instanton in anintersecting brane configuration additional zero modes can only arise from theintersection of the instanton Ξ with D6-branes Πa There are Na [ΞcapΠa]

+ chiralfermionic zero modes λaI and Na [Ξ cap Πa]

minus anti-chiral ones λaJ 2

For its presentation it is useful to introduce the short-hand notation

Φak bk [~xk] = Φak xk1middot Φxk1xk2

middot Φxk2xk3middot middot Φxknminus1xkn

middot Φxkn(k)bk (63)

for the chain-product of open string vertex operators Here we define Φak bk [~0] =Φak bk

To extract the superpotential one can probe it by evaluating an appropriatematter field correlator in the instanton background The CFT allows one tocompute it in physical normalisation which combines the superpotential part Ywith the matter field Kahler metrics like

〈Φa1b1 middot middot ΦaM bM 〉E2minusinst =e

K

2 YΦa1b1ΦaMbMradic

Ka1b1 middot middotKaM bM

(64)

In [14] a general expression for the single E2-instanton contribution to thecharged matter superpotential was proposed involving the evaluation of the fol-lowing zero mode integral over disc and one-loop open string CFT amplitudes

〈Φa1b1 middot middot ΦaM bM 〉E2 =V3gs

intd4x d2θ

sum

conf

proda

(prod[ΞcapΠa]+

i=1 dλia) (prod[ΞcapΠa]minus

i=1 dλi

a

)

exp(minusSE2) exp (Z prime0(E2)) 〈Φa1b1[~x1]〉λa1 λb1

middot middot 〈ΦaLbL[~xL]〉λaLλbL

(65)

For simplicity we do not consider the case that matter fields are also assignedto string loop diagrams The one-loop contributions are annulus diagrams for

2Here we introduced the physical intersection number between two branes Πa cap Πb whichis the sum of positive [Πa capΠb]

+ and negative [Πa cap Πb]minus intersections

17

open strings with one boundary on the E2-instanton and the other boundary onthe various D6-branes and Mobius diagrams with boundary on the E2-instanton

〈1〉1-loop = Z prime0(E2) =

sumb Z

primeA(E2aD6b) + Z primeM(E2aO6) (66)

Here Z prime means that we only sum over the massive open string states in theloop amplitude as the zero modes are taken care of explicitly It was shownthat these instantonic open string loop diagrams are identical to the one-loopthreshold corrections TA(D6aD6b) Diagrammatically we have the intriguingrelation shown in figure 1 and in figure 2 which holds for the even spin structures3

Fa

Fa x

x

D6aaE2 =D6b D6b

Figure 1 Relation between instantonic one-loop amplitudes and corresponding gaugethreshold corrections

Fa

Fa x

x

D6aaE2 =

O6O6

Figure 2 Relation between instantonic Mobius amplitude and corresponding gaugethreshold corrections

The annulus threshold corrections can be computed leading to

ZA(E2aD6b) =

int infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it)

η3(it)ACY

ab [αβ](it) (67)

and the Mobius strip amplitude for the instanton which as we explained mustbe invariant under the orientifold projection yields

ZM(E2aO6) = plusmnint infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it + 1

2

)

η3(it + 1

2

) ACYaa [α

β](it + 1

2

)(68)

3The contribution of the CP-odd Rminus sector is expected to yield corrections to the θ-angle

18

The overall plus sign is for O(1) instantons reflecting the fact that only for thesethe xmicro and θα zero modes survive the orientifold projection Note that up to theargument the Mobius thresholds are ZA(E2aD6a) Therefore for rigid branesthe massless sector reflects the number of four bosonic and two fermionic zeromodes In section 63 we will discuss the number of zero modes if b1(Ξ) gt 0 Allthese stringy threshold corrections are known to be non-holomorphic Thereforeit is not immediately obvious in which sense the expression (65) is meant andhow one can extract the holomorphic superpotential part Y from it

The CFT disc amplitudes in (65) are also not holomorphic but combinenon-holomorphic Kahler potential contributions and holomorphic superpotentialcontributions in the usual way [38 39 40 41]

〈Φab[~x]〉λaλb=

eK

2 YλaΦax1Φx1x2 ΦxN b λbradicKλaaKax1 KxnbKbλb

(69)

=e

K

2 YλabΦab[x]λbradic

Kλaa Kab[x]Kbλb

(610)

Due to the Kaplunovsky-Louis formula (51) the stringy one-loop amplitudes areknown to include the holomorphic Wilsonian part and contributions from wave-function normalisation Applied to the instanton one-loop amplitudes appearingin Z0(E2a) we write

Z0(E2a) = minus8π2 real(f (1)a )minus ba

2ln

(M2

p

micro2

)minus ca

2Ktree (611)

minus ln

(V3gs

)

tree

+sum

b

|IabNb|2

ln[detKab

]tree

where for the brane and instanton configuration in question the coefficients are

ba =sum

b

|IabNb|2

minus 3 ca =sum

b

|IabNb|2

minus 1 (612)

The constant contributions arise from the Mobius amplitude Inserting (69) and(611) in (65) one realises that the Kahler metrics involving an instanton zeromode and a matter field precisely cancel out so that only the matter metricssurvive as required by the general form (64) Moreover the term exp(K2)comes out just right due to the rule that each disc contains precisely two instantonzero modes The holomorphic piece in (64) can therefore be expressed entirelyin terms of other holomorphic quantities like holomorphic Yukawa couplingsthe holomorphic instanton action and the one-loop holomorphic Wilsonian gaugekinetic function on the E2-brane

YΦa1b1ΦaMbM

=sum

conf

signconf exp(minusSE2)tree exp(minusf (1)

a

)

Yλa1bΦa1b1

[~x1] λb1middot middot Yλa1

bΦaLbL[~xL]λbL

(613)

19

This explicitly shows that knowing the tree-level Kahler potentials computingthe matter field correlator in the instanton background up to one-loop level ings is sufficient to deduce the Wilsonian holomorphic instanton generated super-potential Higher order corrections in gs only come from loop corrections to theKahler potentials

63 Instanton corrections to the gauge kinetic functions

So far we have discussed space-time instanton corrections to the superpotentialThese involved one-loop determinants which are given by annulus vacuum di-agrams with at least one E2 boundary These are related to one-loop gaugethreshold corrections to the gauge theory on a D6-brane wrapping the same cycleas the E2 instanton

Now we can ask what other corrections these space-time instantons can induceBy applying S- and T-dualities to the story of world-sheet instanton correctionsin the heterotic string we expect that there can also be E2-instanton corrections

to the holomorphic gauge kinetic functions In the heterotic case similar to thetopological Type II string such corrections arise from string world-sheets of Eulercharacteristic zero ie here from world-sheets with two boundaries Thereforewe expect such corrections to appear for E2-instantons admitting one complexopen string modulus ie those wrapping a three-cycle with Betti number b1(Ξ) =1

Let us start by discussing the instanton zero mode structure for such a cycleFirst let us provide the form of the vertex operators The bosonic fields in the(minus1) ghost picture are

V (minus1)y (z) = y eminusϕ(z)Σh= 1

2q=plusmn1(z) (614)

which before the orientifold projection are accompanied by the two pairs offermionic zero modes

V (minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 1

2(z) (615)

and

V(minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=+ 1

2(z) (616)

Now one has to distinguish two cases depending on how the anti-holomorphicinvolution σ acts on the open string modulus Y

σ y rarr plusmny (617)

In the case that y is invariant under σ called first kind in the following theorientifold projection acts in the same way as for the 4D fields Xmicro ie thetwo bosonic zero modes y and the two fermionic zero modes micro survive In the

20

other case dubbed second kind the bosonic zero mode is projected out andonly the fermionic modulino zero mode micro survives4 Therefore in the absence ofany additional zero modes for instance from E2-D6 intersections the zero modemeasure in any instanton amplitude assumes the following form

intd4x d2θ d2y d2micro eminusSE2 for σ y rarr y (618)

andintd4x d2θ d2micro eminusSE2 for σ y rarr minusy (619)

As an example consider the set-up in figure 3 with σ yi rarr minusyi Here the

x x x2 31

1 2 3y y y

O6

E2

∆x1 ∆ y3

∆x2

Figure 3 Deformations of an instanton which is invariant under the orientifold pro-jection

deformations ∆x12 are of the first kind and ∆y3 is of the second kindNow it is clear that an instanton with precisely one set of fermionic zero

modes of the second kind and no additional zero modes can generate a correctionto the SU(Na) gauge kinetic function The instanton amplitude takes on thefollowing form

〈Fa(p1)Fa(p2)〉E2 =

intd4x d2θ d2micro exp(minusSE2) exp (Z prime

0(E2)) AF 2a(E2 D6a)

where AF 2a(E2 D6a) is the annulus diagram in figure 4 which absorbs all the

appearing fermionic zero modes and where the gauge boson vertex operators inthe (0)-ghost picture have the usual form

V(0)A (z) = ǫmicro (partmicroX(z) + i(p middot ψ)ψmicro(z)) e

ipmiddotX(z) (620)

4By duality this distinction is related to the two kinds of deformations of genus g curvesstudied in [42] The first kind are the curves moving in families ie transversal deformationof the curve The second kind is related to the deformations coming with the genus g of thecurve

21

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

where supersymmetry of course requiressum3

I=1 θIab = 0 Second the holomorphic

gauge kinetic function (on the background considered) only receives correctionsfrom N = 2 open string sectors and the one-loop correction takes on the followingform

f (1)a = minus

sum

b

Nb |IJabIKab |4π2

ln η(i T cI ) I 6= J 6= K 6= I (525)

where the sum only runs over branes b which lie on top of brane a in exactly onetorus denoted by I Therefore the results for the gauge threshold corrections andthe matter field Kahler metrics are consistent both with the non-renormalisationtheorem from section 2 and the Kaplunovsky-Louis formula (51) Clearly itwould be interesting along the lines of [34] to carry out a string amplitude com-putation to fix the free coefficient in the ansatz (511) and see whether our indirectarguments are correct

6 Holomorphic E2-instanton amplitudes

Space-time instantons are given also by D-branes which in this case are EuclideanD2-branes (so-called E2-branes) wrapping three-cycles Ξ in the CalabindashYau sothat they are point-like in four-dimensional Minkowski space Such instantonscan contribute to the holomorphic superpotential and gauge kinetic functions onlyif they preserve half of the N = 1 supersymmetry This means that the instantonmeasure must contain a factor d4x d2θ Let us first clarify an important aspectof this half-BPS condition In the second part of this section we then revisitthe computation of contributions of such instantons to the superpotential andalso clarify some issues concerned with the appearing one-loop determinants Inthe third and fourth part we investigate under which conditions such stringinstantons can also contribute to the gauge kinetic functions and FI-terms

61 Half-BPS instantons

As has been explained in [24 18 20] just wrapping an E2-instanton around arigid sLag three-cycle in the Calabi-Yau gives four bosonic and four fermioniczero modes The vertex operators for the latter are

V(minus12)θ (z) = θα e

minusϕ(z)2 (z)Sα(z) Σh= 3

8q= 3

2(z) (61)

and

V(minus12)

θ(z) = θα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 3

2(z) (62)

Therefore if the instanton is not invariant under the orientifold projection onestill has four instead of the desired two fermionic zero modes Thus only by

16

placing the E2-brane in a position invariant under Ωσ does one have a chanceto get rid of the two additional zero modes θ For so called O(n) instantons onecan see that the zero modes xmicro θ are symmetrised and the mode θ gets anti-symmetrised For the opposite projection ie for USp(2n) instantons the zeromodes xmicro θ are anti-symmetrised and the mode θ gets symmetrised Thereforeone can only get the simple d4x d2θ instanton measure for a single O(1) instanton

62 Superpotential contributions

In order to contribute to the superpotential we also require that there do notarise any further zero modes from E2-E2 open strings so that the three-cycleΞ should be rigid ie b1(Ξ) = 0 Therefore considering an E2-instanton in anintersecting brane configuration additional zero modes can only arise from theintersection of the instanton Ξ with D6-branes Πa There are Na [ΞcapΠa]

+ chiralfermionic zero modes λaI and Na [Ξ cap Πa]

minus anti-chiral ones λaJ 2

For its presentation it is useful to introduce the short-hand notation

Φak bk [~xk] = Φak xk1middot Φxk1xk2

middot Φxk2xk3middot middot Φxknminus1xkn

middot Φxkn(k)bk (63)

for the chain-product of open string vertex operators Here we define Φak bk [~0] =Φak bk

To extract the superpotential one can probe it by evaluating an appropriatematter field correlator in the instanton background The CFT allows one tocompute it in physical normalisation which combines the superpotential part Ywith the matter field Kahler metrics like

〈Φa1b1 middot middot ΦaM bM 〉E2minusinst =e

K

2 YΦa1b1ΦaMbMradic

Ka1b1 middot middotKaM bM

(64)

In [14] a general expression for the single E2-instanton contribution to thecharged matter superpotential was proposed involving the evaluation of the fol-lowing zero mode integral over disc and one-loop open string CFT amplitudes

〈Φa1b1 middot middot ΦaM bM 〉E2 =V3gs

intd4x d2θ

sum

conf

proda

(prod[ΞcapΠa]+

i=1 dλia) (prod[ΞcapΠa]minus

i=1 dλi

a

)

exp(minusSE2) exp (Z prime0(E2)) 〈Φa1b1[~x1]〉λa1 λb1

middot middot 〈ΦaLbL[~xL]〉λaLλbL

(65)

For simplicity we do not consider the case that matter fields are also assignedto string loop diagrams The one-loop contributions are annulus diagrams for

2Here we introduced the physical intersection number between two branes Πa cap Πb whichis the sum of positive [Πa capΠb]

+ and negative [Πa cap Πb]minus intersections

17

open strings with one boundary on the E2-instanton and the other boundary onthe various D6-branes and Mobius diagrams with boundary on the E2-instanton

〈1〉1-loop = Z prime0(E2) =

sumb Z

primeA(E2aD6b) + Z primeM(E2aO6) (66)

Here Z prime means that we only sum over the massive open string states in theloop amplitude as the zero modes are taken care of explicitly It was shownthat these instantonic open string loop diagrams are identical to the one-loopthreshold corrections TA(D6aD6b) Diagrammatically we have the intriguingrelation shown in figure 1 and in figure 2 which holds for the even spin structures3

Fa

Fa x

x

D6aaE2 =D6b D6b

Figure 1 Relation between instantonic one-loop amplitudes and corresponding gaugethreshold corrections

Fa

Fa x

x

D6aaE2 =

O6O6

Figure 2 Relation between instantonic Mobius amplitude and corresponding gaugethreshold corrections

The annulus threshold corrections can be computed leading to

ZA(E2aD6b) =

int infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it)

η3(it)ACY

ab [αβ](it) (67)

and the Mobius strip amplitude for the instanton which as we explained mustbe invariant under the orientifold projection yields

ZM(E2aO6) = plusmnint infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it + 1

2

)

η3(it + 1

2

) ACYaa [α

β](it + 1

2

)(68)

3The contribution of the CP-odd Rminus sector is expected to yield corrections to the θ-angle

18

The overall plus sign is for O(1) instantons reflecting the fact that only for thesethe xmicro and θα zero modes survive the orientifold projection Note that up to theargument the Mobius thresholds are ZA(E2aD6a) Therefore for rigid branesthe massless sector reflects the number of four bosonic and two fermionic zeromodes In section 63 we will discuss the number of zero modes if b1(Ξ) gt 0 Allthese stringy threshold corrections are known to be non-holomorphic Thereforeit is not immediately obvious in which sense the expression (65) is meant andhow one can extract the holomorphic superpotential part Y from it

The CFT disc amplitudes in (65) are also not holomorphic but combinenon-holomorphic Kahler potential contributions and holomorphic superpotentialcontributions in the usual way [38 39 40 41]

〈Φab[~x]〉λaλb=

eK

2 YλaΦax1Φx1x2 ΦxN b λbradicKλaaKax1 KxnbKbλb

(69)

=e

K

2 YλabΦab[x]λbradic

Kλaa Kab[x]Kbλb

(610)

Due to the Kaplunovsky-Louis formula (51) the stringy one-loop amplitudes areknown to include the holomorphic Wilsonian part and contributions from wave-function normalisation Applied to the instanton one-loop amplitudes appearingin Z0(E2a) we write

Z0(E2a) = minus8π2 real(f (1)a )minus ba

2ln

(M2

p

micro2

)minus ca

2Ktree (611)

minus ln

(V3gs

)

tree

+sum

b

|IabNb|2

ln[detKab

]tree

where for the brane and instanton configuration in question the coefficients are

ba =sum

b

|IabNb|2

minus 3 ca =sum

b

|IabNb|2

minus 1 (612)

The constant contributions arise from the Mobius amplitude Inserting (69) and(611) in (65) one realises that the Kahler metrics involving an instanton zeromode and a matter field precisely cancel out so that only the matter metricssurvive as required by the general form (64) Moreover the term exp(K2)comes out just right due to the rule that each disc contains precisely two instantonzero modes The holomorphic piece in (64) can therefore be expressed entirelyin terms of other holomorphic quantities like holomorphic Yukawa couplingsthe holomorphic instanton action and the one-loop holomorphic Wilsonian gaugekinetic function on the E2-brane

YΦa1b1ΦaMbM

=sum

conf

signconf exp(minusSE2)tree exp(minusf (1)

a

)

Yλa1bΦa1b1

[~x1] λb1middot middot Yλa1

bΦaLbL[~xL]λbL

(613)

19

This explicitly shows that knowing the tree-level Kahler potentials computingthe matter field correlator in the instanton background up to one-loop level ings is sufficient to deduce the Wilsonian holomorphic instanton generated super-potential Higher order corrections in gs only come from loop corrections to theKahler potentials

63 Instanton corrections to the gauge kinetic functions

So far we have discussed space-time instanton corrections to the superpotentialThese involved one-loop determinants which are given by annulus vacuum di-agrams with at least one E2 boundary These are related to one-loop gaugethreshold corrections to the gauge theory on a D6-brane wrapping the same cycleas the E2 instanton

Now we can ask what other corrections these space-time instantons can induceBy applying S- and T-dualities to the story of world-sheet instanton correctionsin the heterotic string we expect that there can also be E2-instanton corrections

to the holomorphic gauge kinetic functions In the heterotic case similar to thetopological Type II string such corrections arise from string world-sheets of Eulercharacteristic zero ie here from world-sheets with two boundaries Thereforewe expect such corrections to appear for E2-instantons admitting one complexopen string modulus ie those wrapping a three-cycle with Betti number b1(Ξ) =1

Let us start by discussing the instanton zero mode structure for such a cycleFirst let us provide the form of the vertex operators The bosonic fields in the(minus1) ghost picture are

V (minus1)y (z) = y eminusϕ(z)Σh= 1

2q=plusmn1(z) (614)

which before the orientifold projection are accompanied by the two pairs offermionic zero modes

V (minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 1

2(z) (615)

and

V(minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=+ 1

2(z) (616)

Now one has to distinguish two cases depending on how the anti-holomorphicinvolution σ acts on the open string modulus Y

σ y rarr plusmny (617)

In the case that y is invariant under σ called first kind in the following theorientifold projection acts in the same way as for the 4D fields Xmicro ie thetwo bosonic zero modes y and the two fermionic zero modes micro survive In the

20

other case dubbed second kind the bosonic zero mode is projected out andonly the fermionic modulino zero mode micro survives4 Therefore in the absence ofany additional zero modes for instance from E2-D6 intersections the zero modemeasure in any instanton amplitude assumes the following form

intd4x d2θ d2y d2micro eminusSE2 for σ y rarr y (618)

andintd4x d2θ d2micro eminusSE2 for σ y rarr minusy (619)

As an example consider the set-up in figure 3 with σ yi rarr minusyi Here the

x x x2 31

1 2 3y y y

O6

E2

∆x1 ∆ y3

∆x2

Figure 3 Deformations of an instanton which is invariant under the orientifold pro-jection

deformations ∆x12 are of the first kind and ∆y3 is of the second kindNow it is clear that an instanton with precisely one set of fermionic zero

modes of the second kind and no additional zero modes can generate a correctionto the SU(Na) gauge kinetic function The instanton amplitude takes on thefollowing form

〈Fa(p1)Fa(p2)〉E2 =

intd4x d2θ d2micro exp(minusSE2) exp (Z prime

0(E2)) AF 2a(E2 D6a)

where AF 2a(E2 D6a) is the annulus diagram in figure 4 which absorbs all the

appearing fermionic zero modes and where the gauge boson vertex operators inthe (0)-ghost picture have the usual form

V(0)A (z) = ǫmicro (partmicroX(z) + i(p middot ψ)ψmicro(z)) e

ipmiddotX(z) (620)

4By duality this distinction is related to the two kinds of deformations of genus g curvesstudied in [42] The first kind are the curves moving in families ie transversal deformationof the curve The second kind is related to the deformations coming with the genus g of thecurve

21

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

placing the E2-brane in a position invariant under Ωσ does one have a chanceto get rid of the two additional zero modes θ For so called O(n) instantons onecan see that the zero modes xmicro θ are symmetrised and the mode θ gets anti-symmetrised For the opposite projection ie for USp(2n) instantons the zeromodes xmicro θ are anti-symmetrised and the mode θ gets symmetrised Thereforeone can only get the simple d4x d2θ instanton measure for a single O(1) instanton

62 Superpotential contributions

In order to contribute to the superpotential we also require that there do notarise any further zero modes from E2-E2 open strings so that the three-cycleΞ should be rigid ie b1(Ξ) = 0 Therefore considering an E2-instanton in anintersecting brane configuration additional zero modes can only arise from theintersection of the instanton Ξ with D6-branes Πa There are Na [ΞcapΠa]

+ chiralfermionic zero modes λaI and Na [Ξ cap Πa]

minus anti-chiral ones λaJ 2

For its presentation it is useful to introduce the short-hand notation

Φak bk [~xk] = Φak xk1middot Φxk1xk2

middot Φxk2xk3middot middot Φxknminus1xkn

middot Φxkn(k)bk (63)

for the chain-product of open string vertex operators Here we define Φak bk [~0] =Φak bk

To extract the superpotential one can probe it by evaluating an appropriatematter field correlator in the instanton background The CFT allows one tocompute it in physical normalisation which combines the superpotential part Ywith the matter field Kahler metrics like

〈Φa1b1 middot middot ΦaM bM 〉E2minusinst =e

K

2 YΦa1b1ΦaMbMradic

Ka1b1 middot middotKaM bM

(64)

In [14] a general expression for the single E2-instanton contribution to thecharged matter superpotential was proposed involving the evaluation of the fol-lowing zero mode integral over disc and one-loop open string CFT amplitudes

〈Φa1b1 middot middot ΦaM bM 〉E2 =V3gs

intd4x d2θ

sum

conf

proda

(prod[ΞcapΠa]+

i=1 dλia) (prod[ΞcapΠa]minus

i=1 dλi

a

)

exp(minusSE2) exp (Z prime0(E2)) 〈Φa1b1[~x1]〉λa1 λb1

middot middot 〈ΦaLbL[~xL]〉λaLλbL

(65)

For simplicity we do not consider the case that matter fields are also assignedto string loop diagrams The one-loop contributions are annulus diagrams for

2Here we introduced the physical intersection number between two branes Πa cap Πb whichis the sum of positive [Πa capΠb]

+ and negative [Πa cap Πb]minus intersections

17

open strings with one boundary on the E2-instanton and the other boundary onthe various D6-branes and Mobius diagrams with boundary on the E2-instanton

〈1〉1-loop = Z prime0(E2) =

sumb Z

primeA(E2aD6b) + Z primeM(E2aO6) (66)

Here Z prime means that we only sum over the massive open string states in theloop amplitude as the zero modes are taken care of explicitly It was shownthat these instantonic open string loop diagrams are identical to the one-loopthreshold corrections TA(D6aD6b) Diagrammatically we have the intriguingrelation shown in figure 1 and in figure 2 which holds for the even spin structures3

Fa

Fa x

x

D6aaE2 =D6b D6b

Figure 1 Relation between instantonic one-loop amplitudes and corresponding gaugethreshold corrections

Fa

Fa x

x

D6aaE2 =

O6O6

Figure 2 Relation between instantonic Mobius amplitude and corresponding gaugethreshold corrections

The annulus threshold corrections can be computed leading to

ZA(E2aD6b) =

int infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it)

η3(it)ACY

ab [αβ](it) (67)

and the Mobius strip amplitude for the instanton which as we explained mustbe invariant under the orientifold projection yields

ZM(E2aO6) = plusmnint infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it + 1

2

)

η3(it + 1

2

) ACYaa [α

β](it + 1

2

)(68)

3The contribution of the CP-odd Rminus sector is expected to yield corrections to the θ-angle

18

The overall plus sign is for O(1) instantons reflecting the fact that only for thesethe xmicro and θα zero modes survive the orientifold projection Note that up to theargument the Mobius thresholds are ZA(E2aD6a) Therefore for rigid branesthe massless sector reflects the number of four bosonic and two fermionic zeromodes In section 63 we will discuss the number of zero modes if b1(Ξ) gt 0 Allthese stringy threshold corrections are known to be non-holomorphic Thereforeit is not immediately obvious in which sense the expression (65) is meant andhow one can extract the holomorphic superpotential part Y from it

The CFT disc amplitudes in (65) are also not holomorphic but combinenon-holomorphic Kahler potential contributions and holomorphic superpotentialcontributions in the usual way [38 39 40 41]

〈Φab[~x]〉λaλb=

eK

2 YλaΦax1Φx1x2 ΦxN b λbradicKλaaKax1 KxnbKbλb

(69)

=e

K

2 YλabΦab[x]λbradic

Kλaa Kab[x]Kbλb

(610)

Due to the Kaplunovsky-Louis formula (51) the stringy one-loop amplitudes areknown to include the holomorphic Wilsonian part and contributions from wave-function normalisation Applied to the instanton one-loop amplitudes appearingin Z0(E2a) we write

Z0(E2a) = minus8π2 real(f (1)a )minus ba

2ln

(M2

p

micro2

)minus ca

2Ktree (611)

minus ln

(V3gs

)

tree

+sum

b

|IabNb|2

ln[detKab

]tree

where for the brane and instanton configuration in question the coefficients are

ba =sum

b

|IabNb|2

minus 3 ca =sum

b

|IabNb|2

minus 1 (612)

The constant contributions arise from the Mobius amplitude Inserting (69) and(611) in (65) one realises that the Kahler metrics involving an instanton zeromode and a matter field precisely cancel out so that only the matter metricssurvive as required by the general form (64) Moreover the term exp(K2)comes out just right due to the rule that each disc contains precisely two instantonzero modes The holomorphic piece in (64) can therefore be expressed entirelyin terms of other holomorphic quantities like holomorphic Yukawa couplingsthe holomorphic instanton action and the one-loop holomorphic Wilsonian gaugekinetic function on the E2-brane

YΦa1b1ΦaMbM

=sum

conf

signconf exp(minusSE2)tree exp(minusf (1)

a

)

Yλa1bΦa1b1

[~x1] λb1middot middot Yλa1

bΦaLbL[~xL]λbL

(613)

19

This explicitly shows that knowing the tree-level Kahler potentials computingthe matter field correlator in the instanton background up to one-loop level ings is sufficient to deduce the Wilsonian holomorphic instanton generated super-potential Higher order corrections in gs only come from loop corrections to theKahler potentials

63 Instanton corrections to the gauge kinetic functions

So far we have discussed space-time instanton corrections to the superpotentialThese involved one-loop determinants which are given by annulus vacuum di-agrams with at least one E2 boundary These are related to one-loop gaugethreshold corrections to the gauge theory on a D6-brane wrapping the same cycleas the E2 instanton

Now we can ask what other corrections these space-time instantons can induceBy applying S- and T-dualities to the story of world-sheet instanton correctionsin the heterotic string we expect that there can also be E2-instanton corrections

to the holomorphic gauge kinetic functions In the heterotic case similar to thetopological Type II string such corrections arise from string world-sheets of Eulercharacteristic zero ie here from world-sheets with two boundaries Thereforewe expect such corrections to appear for E2-instantons admitting one complexopen string modulus ie those wrapping a three-cycle with Betti number b1(Ξ) =1

Let us start by discussing the instanton zero mode structure for such a cycleFirst let us provide the form of the vertex operators The bosonic fields in the(minus1) ghost picture are

V (minus1)y (z) = y eminusϕ(z)Σh= 1

2q=plusmn1(z) (614)

which before the orientifold projection are accompanied by the two pairs offermionic zero modes

V (minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 1

2(z) (615)

and

V(minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=+ 1

2(z) (616)

Now one has to distinguish two cases depending on how the anti-holomorphicinvolution σ acts on the open string modulus Y

σ y rarr plusmny (617)

In the case that y is invariant under σ called first kind in the following theorientifold projection acts in the same way as for the 4D fields Xmicro ie thetwo bosonic zero modes y and the two fermionic zero modes micro survive In the

20

other case dubbed second kind the bosonic zero mode is projected out andonly the fermionic modulino zero mode micro survives4 Therefore in the absence ofany additional zero modes for instance from E2-D6 intersections the zero modemeasure in any instanton amplitude assumes the following form

intd4x d2θ d2y d2micro eminusSE2 for σ y rarr y (618)

andintd4x d2θ d2micro eminusSE2 for σ y rarr minusy (619)

As an example consider the set-up in figure 3 with σ yi rarr minusyi Here the

x x x2 31

1 2 3y y y

O6

E2

∆x1 ∆ y3

∆x2

Figure 3 Deformations of an instanton which is invariant under the orientifold pro-jection

deformations ∆x12 are of the first kind and ∆y3 is of the second kindNow it is clear that an instanton with precisely one set of fermionic zero

modes of the second kind and no additional zero modes can generate a correctionto the SU(Na) gauge kinetic function The instanton amplitude takes on thefollowing form

〈Fa(p1)Fa(p2)〉E2 =

intd4x d2θ d2micro exp(minusSE2) exp (Z prime

0(E2)) AF 2a(E2 D6a)

where AF 2a(E2 D6a) is the annulus diagram in figure 4 which absorbs all the

appearing fermionic zero modes and where the gauge boson vertex operators inthe (0)-ghost picture have the usual form

V(0)A (z) = ǫmicro (partmicroX(z) + i(p middot ψ)ψmicro(z)) e

ipmiddotX(z) (620)

4By duality this distinction is related to the two kinds of deformations of genus g curvesstudied in [42] The first kind are the curves moving in families ie transversal deformationof the curve The second kind is related to the deformations coming with the genus g of thecurve

21

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

open strings with one boundary on the E2-instanton and the other boundary onthe various D6-branes and Mobius diagrams with boundary on the E2-instanton

〈1〉1-loop = Z prime0(E2) =

sumb Z

primeA(E2aD6b) + Z primeM(E2aO6) (66)

Here Z prime means that we only sum over the massive open string states in theloop amplitude as the zero modes are taken care of explicitly It was shownthat these instantonic open string loop diagrams are identical to the one-loopthreshold corrections TA(D6aD6b) Diagrammatically we have the intriguingrelation shown in figure 1 and in figure 2 which holds for the even spin structures3

Fa

Fa x

x

D6aaE2 =D6b D6b

Figure 1 Relation between instantonic one-loop amplitudes and corresponding gaugethreshold corrections

Fa

Fa x

x

D6aaE2 =

O6O6

Figure 2 Relation between instantonic Mobius amplitude and corresponding gaugethreshold corrections

The annulus threshold corrections can be computed leading to

ZA(E2aD6b) =

int infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it)

η3(it)ACY

ab [αβ](it) (67)

and the Mobius strip amplitude for the instanton which as we explained mustbe invariant under the orientifold projection yields

ZM(E2aO6) = plusmnint infin

0

dt

t

sum

αβ 6=( 12 12)

(minus1)2(α+β)ϑprimeprime[α

β](it + 1

2

)

η3(it + 1

2

) ACYaa [α

β](it + 1

2

)(68)

3The contribution of the CP-odd Rminus sector is expected to yield corrections to the θ-angle

18

The overall plus sign is for O(1) instantons reflecting the fact that only for thesethe xmicro and θα zero modes survive the orientifold projection Note that up to theargument the Mobius thresholds are ZA(E2aD6a) Therefore for rigid branesthe massless sector reflects the number of four bosonic and two fermionic zeromodes In section 63 we will discuss the number of zero modes if b1(Ξ) gt 0 Allthese stringy threshold corrections are known to be non-holomorphic Thereforeit is not immediately obvious in which sense the expression (65) is meant andhow one can extract the holomorphic superpotential part Y from it

The CFT disc amplitudes in (65) are also not holomorphic but combinenon-holomorphic Kahler potential contributions and holomorphic superpotentialcontributions in the usual way [38 39 40 41]

〈Φab[~x]〉λaλb=

eK

2 YλaΦax1Φx1x2 ΦxN b λbradicKλaaKax1 KxnbKbλb

(69)

=e

K

2 YλabΦab[x]λbradic

Kλaa Kab[x]Kbλb

(610)

Due to the Kaplunovsky-Louis formula (51) the stringy one-loop amplitudes areknown to include the holomorphic Wilsonian part and contributions from wave-function normalisation Applied to the instanton one-loop amplitudes appearingin Z0(E2a) we write

Z0(E2a) = minus8π2 real(f (1)a )minus ba

2ln

(M2

p

micro2

)minus ca

2Ktree (611)

minus ln

(V3gs

)

tree

+sum

b

|IabNb|2

ln[detKab

]tree

where for the brane and instanton configuration in question the coefficients are

ba =sum

b

|IabNb|2

minus 3 ca =sum

b

|IabNb|2

minus 1 (612)

The constant contributions arise from the Mobius amplitude Inserting (69) and(611) in (65) one realises that the Kahler metrics involving an instanton zeromode and a matter field precisely cancel out so that only the matter metricssurvive as required by the general form (64) Moreover the term exp(K2)comes out just right due to the rule that each disc contains precisely two instantonzero modes The holomorphic piece in (64) can therefore be expressed entirelyin terms of other holomorphic quantities like holomorphic Yukawa couplingsthe holomorphic instanton action and the one-loop holomorphic Wilsonian gaugekinetic function on the E2-brane

YΦa1b1ΦaMbM

=sum

conf

signconf exp(minusSE2)tree exp(minusf (1)

a

)

Yλa1bΦa1b1

[~x1] λb1middot middot Yλa1

bΦaLbL[~xL]λbL

(613)

19

This explicitly shows that knowing the tree-level Kahler potentials computingthe matter field correlator in the instanton background up to one-loop level ings is sufficient to deduce the Wilsonian holomorphic instanton generated super-potential Higher order corrections in gs only come from loop corrections to theKahler potentials

63 Instanton corrections to the gauge kinetic functions

So far we have discussed space-time instanton corrections to the superpotentialThese involved one-loop determinants which are given by annulus vacuum di-agrams with at least one E2 boundary These are related to one-loop gaugethreshold corrections to the gauge theory on a D6-brane wrapping the same cycleas the E2 instanton

Now we can ask what other corrections these space-time instantons can induceBy applying S- and T-dualities to the story of world-sheet instanton correctionsin the heterotic string we expect that there can also be E2-instanton corrections

to the holomorphic gauge kinetic functions In the heterotic case similar to thetopological Type II string such corrections arise from string world-sheets of Eulercharacteristic zero ie here from world-sheets with two boundaries Thereforewe expect such corrections to appear for E2-instantons admitting one complexopen string modulus ie those wrapping a three-cycle with Betti number b1(Ξ) =1

Let us start by discussing the instanton zero mode structure for such a cycleFirst let us provide the form of the vertex operators The bosonic fields in the(minus1) ghost picture are

V (minus1)y (z) = y eminusϕ(z)Σh= 1

2q=plusmn1(z) (614)

which before the orientifold projection are accompanied by the two pairs offermionic zero modes

V (minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 1

2(z) (615)

and

V(minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=+ 1

2(z) (616)

Now one has to distinguish two cases depending on how the anti-holomorphicinvolution σ acts on the open string modulus Y

σ y rarr plusmny (617)

In the case that y is invariant under σ called first kind in the following theorientifold projection acts in the same way as for the 4D fields Xmicro ie thetwo bosonic zero modes y and the two fermionic zero modes micro survive In the

20

other case dubbed second kind the bosonic zero mode is projected out andonly the fermionic modulino zero mode micro survives4 Therefore in the absence ofany additional zero modes for instance from E2-D6 intersections the zero modemeasure in any instanton amplitude assumes the following form

intd4x d2θ d2y d2micro eminusSE2 for σ y rarr y (618)

andintd4x d2θ d2micro eminusSE2 for σ y rarr minusy (619)

As an example consider the set-up in figure 3 with σ yi rarr minusyi Here the

x x x2 31

1 2 3y y y

O6

E2

∆x1 ∆ y3

∆x2

Figure 3 Deformations of an instanton which is invariant under the orientifold pro-jection

deformations ∆x12 are of the first kind and ∆y3 is of the second kindNow it is clear that an instanton with precisely one set of fermionic zero

modes of the second kind and no additional zero modes can generate a correctionto the SU(Na) gauge kinetic function The instanton amplitude takes on thefollowing form

〈Fa(p1)Fa(p2)〉E2 =

intd4x d2θ d2micro exp(minusSE2) exp (Z prime

0(E2)) AF 2a(E2 D6a)

where AF 2a(E2 D6a) is the annulus diagram in figure 4 which absorbs all the

appearing fermionic zero modes and where the gauge boson vertex operators inthe (0)-ghost picture have the usual form

V(0)A (z) = ǫmicro (partmicroX(z) + i(p middot ψ)ψmicro(z)) e

ipmiddotX(z) (620)

4By duality this distinction is related to the two kinds of deformations of genus g curvesstudied in [42] The first kind are the curves moving in families ie transversal deformationof the curve The second kind is related to the deformations coming with the genus g of thecurve

21

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

The overall plus sign is for O(1) instantons reflecting the fact that only for thesethe xmicro and θα zero modes survive the orientifold projection Note that up to theargument the Mobius thresholds are ZA(E2aD6a) Therefore for rigid branesthe massless sector reflects the number of four bosonic and two fermionic zeromodes In section 63 we will discuss the number of zero modes if b1(Ξ) gt 0 Allthese stringy threshold corrections are known to be non-holomorphic Thereforeit is not immediately obvious in which sense the expression (65) is meant andhow one can extract the holomorphic superpotential part Y from it

The CFT disc amplitudes in (65) are also not holomorphic but combinenon-holomorphic Kahler potential contributions and holomorphic superpotentialcontributions in the usual way [38 39 40 41]

〈Φab[~x]〉λaλb=

eK

2 YλaΦax1Φx1x2 ΦxN b λbradicKλaaKax1 KxnbKbλb

(69)

=e

K

2 YλabΦab[x]λbradic

Kλaa Kab[x]Kbλb

(610)

Due to the Kaplunovsky-Louis formula (51) the stringy one-loop amplitudes areknown to include the holomorphic Wilsonian part and contributions from wave-function normalisation Applied to the instanton one-loop amplitudes appearingin Z0(E2a) we write

Z0(E2a) = minus8π2 real(f (1)a )minus ba

2ln

(M2

p

micro2

)minus ca

2Ktree (611)

minus ln

(V3gs

)

tree

+sum

b

|IabNb|2

ln[detKab

]tree

where for the brane and instanton configuration in question the coefficients are

ba =sum

b

|IabNb|2

minus 3 ca =sum

b

|IabNb|2

minus 1 (612)

The constant contributions arise from the Mobius amplitude Inserting (69) and(611) in (65) one realises that the Kahler metrics involving an instanton zeromode and a matter field precisely cancel out so that only the matter metricssurvive as required by the general form (64) Moreover the term exp(K2)comes out just right due to the rule that each disc contains precisely two instantonzero modes The holomorphic piece in (64) can therefore be expressed entirelyin terms of other holomorphic quantities like holomorphic Yukawa couplingsthe holomorphic instanton action and the one-loop holomorphic Wilsonian gaugekinetic function on the E2-brane

YΦa1b1ΦaMbM

=sum

conf

signconf exp(minusSE2)tree exp(minusf (1)

a

)

Yλa1bΦa1b1

[~x1] λb1middot middot Yλa1

bΦaLbL[~xL]λbL

(613)

19

This explicitly shows that knowing the tree-level Kahler potentials computingthe matter field correlator in the instanton background up to one-loop level ings is sufficient to deduce the Wilsonian holomorphic instanton generated super-potential Higher order corrections in gs only come from loop corrections to theKahler potentials

63 Instanton corrections to the gauge kinetic functions

So far we have discussed space-time instanton corrections to the superpotentialThese involved one-loop determinants which are given by annulus vacuum di-agrams with at least one E2 boundary These are related to one-loop gaugethreshold corrections to the gauge theory on a D6-brane wrapping the same cycleas the E2 instanton

Now we can ask what other corrections these space-time instantons can induceBy applying S- and T-dualities to the story of world-sheet instanton correctionsin the heterotic string we expect that there can also be E2-instanton corrections

to the holomorphic gauge kinetic functions In the heterotic case similar to thetopological Type II string such corrections arise from string world-sheets of Eulercharacteristic zero ie here from world-sheets with two boundaries Thereforewe expect such corrections to appear for E2-instantons admitting one complexopen string modulus ie those wrapping a three-cycle with Betti number b1(Ξ) =1

Let us start by discussing the instanton zero mode structure for such a cycleFirst let us provide the form of the vertex operators The bosonic fields in the(minus1) ghost picture are

V (minus1)y (z) = y eminusϕ(z)Σh= 1

2q=plusmn1(z) (614)

which before the orientifold projection are accompanied by the two pairs offermionic zero modes

V (minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 1

2(z) (615)

and

V(minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=+ 1

2(z) (616)

Now one has to distinguish two cases depending on how the anti-holomorphicinvolution σ acts on the open string modulus Y

σ y rarr plusmny (617)

In the case that y is invariant under σ called first kind in the following theorientifold projection acts in the same way as for the 4D fields Xmicro ie thetwo bosonic zero modes y and the two fermionic zero modes micro survive In the

20

other case dubbed second kind the bosonic zero mode is projected out andonly the fermionic modulino zero mode micro survives4 Therefore in the absence ofany additional zero modes for instance from E2-D6 intersections the zero modemeasure in any instanton amplitude assumes the following form

intd4x d2θ d2y d2micro eminusSE2 for σ y rarr y (618)

andintd4x d2θ d2micro eminusSE2 for σ y rarr minusy (619)

As an example consider the set-up in figure 3 with σ yi rarr minusyi Here the

x x x2 31

1 2 3y y y

O6

E2

∆x1 ∆ y3

∆x2

Figure 3 Deformations of an instanton which is invariant under the orientifold pro-jection

deformations ∆x12 are of the first kind and ∆y3 is of the second kindNow it is clear that an instanton with precisely one set of fermionic zero

modes of the second kind and no additional zero modes can generate a correctionto the SU(Na) gauge kinetic function The instanton amplitude takes on thefollowing form

〈Fa(p1)Fa(p2)〉E2 =

intd4x d2θ d2micro exp(minusSE2) exp (Z prime

0(E2)) AF 2a(E2 D6a)

where AF 2a(E2 D6a) is the annulus diagram in figure 4 which absorbs all the

appearing fermionic zero modes and where the gauge boson vertex operators inthe (0)-ghost picture have the usual form

V(0)A (z) = ǫmicro (partmicroX(z) + i(p middot ψ)ψmicro(z)) e

ipmiddotX(z) (620)

4By duality this distinction is related to the two kinds of deformations of genus g curvesstudied in [42] The first kind are the curves moving in families ie transversal deformationof the curve The second kind is related to the deformations coming with the genus g of thecurve

21

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

This explicitly shows that knowing the tree-level Kahler potentials computingthe matter field correlator in the instanton background up to one-loop level ings is sufficient to deduce the Wilsonian holomorphic instanton generated super-potential Higher order corrections in gs only come from loop corrections to theKahler potentials

63 Instanton corrections to the gauge kinetic functions

So far we have discussed space-time instanton corrections to the superpotentialThese involved one-loop determinants which are given by annulus vacuum di-agrams with at least one E2 boundary These are related to one-loop gaugethreshold corrections to the gauge theory on a D6-brane wrapping the same cycleas the E2 instanton

Now we can ask what other corrections these space-time instantons can induceBy applying S- and T-dualities to the story of world-sheet instanton correctionsin the heterotic string we expect that there can also be E2-instanton corrections

to the holomorphic gauge kinetic functions In the heterotic case similar to thetopological Type II string such corrections arise from string world-sheets of Eulercharacteristic zero ie here from world-sheets with two boundaries Thereforewe expect such corrections to appear for E2-instantons admitting one complexopen string modulus ie those wrapping a three-cycle with Betti number b1(Ξ) =1

Let us start by discussing the instanton zero mode structure for such a cycleFirst let us provide the form of the vertex operators The bosonic fields in the(minus1) ghost picture are

V (minus1)y (z) = y eminusϕ(z)Σh= 1

2q=plusmn1(z) (614)

which before the orientifold projection are accompanied by the two pairs offermionic zero modes

V (minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=minus 1

2(z) (615)

and

V(minus12)micro (z) = microα e

minusϕ(z)2 Sα(z) Σh= 3

8q=+ 1

2(z) (616)

Now one has to distinguish two cases depending on how the anti-holomorphicinvolution σ acts on the open string modulus Y

σ y rarr plusmny (617)

In the case that y is invariant under σ called first kind in the following theorientifold projection acts in the same way as for the 4D fields Xmicro ie thetwo bosonic zero modes y and the two fermionic zero modes micro survive In the

20

other case dubbed second kind the bosonic zero mode is projected out andonly the fermionic modulino zero mode micro survives4 Therefore in the absence ofany additional zero modes for instance from E2-D6 intersections the zero modemeasure in any instanton amplitude assumes the following form

intd4x d2θ d2y d2micro eminusSE2 for σ y rarr y (618)

andintd4x d2θ d2micro eminusSE2 for σ y rarr minusy (619)

As an example consider the set-up in figure 3 with σ yi rarr minusyi Here the

x x x2 31

1 2 3y y y

O6

E2

∆x1 ∆ y3

∆x2

Figure 3 Deformations of an instanton which is invariant under the orientifold pro-jection

deformations ∆x12 are of the first kind and ∆y3 is of the second kindNow it is clear that an instanton with precisely one set of fermionic zero

modes of the second kind and no additional zero modes can generate a correctionto the SU(Na) gauge kinetic function The instanton amplitude takes on thefollowing form

〈Fa(p1)Fa(p2)〉E2 =

intd4x d2θ d2micro exp(minusSE2) exp (Z prime

0(E2)) AF 2a(E2 D6a)

where AF 2a(E2 D6a) is the annulus diagram in figure 4 which absorbs all the

appearing fermionic zero modes and where the gauge boson vertex operators inthe (0)-ghost picture have the usual form

V(0)A (z) = ǫmicro (partmicroX(z) + i(p middot ψ)ψmicro(z)) e

ipmiddotX(z) (620)

4By duality this distinction is related to the two kinds of deformations of genus g curvesstudied in [42] The first kind are the curves moving in families ie transversal deformationof the curve The second kind is related to the deformations coming with the genus g of thecurve

21

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

other case dubbed second kind the bosonic zero mode is projected out andonly the fermionic modulino zero mode micro survives4 Therefore in the absence ofany additional zero modes for instance from E2-D6 intersections the zero modemeasure in any instanton amplitude assumes the following form

intd4x d2θ d2y d2micro eminusSE2 for σ y rarr y (618)

andintd4x d2θ d2micro eminusSE2 for σ y rarr minusy (619)

As an example consider the set-up in figure 3 with σ yi rarr minusyi Here the

x x x2 31

1 2 3y y y

O6

E2

∆x1 ∆ y3

∆x2

Figure 3 Deformations of an instanton which is invariant under the orientifold pro-jection

deformations ∆x12 are of the first kind and ∆y3 is of the second kindNow it is clear that an instanton with precisely one set of fermionic zero

modes of the second kind and no additional zero modes can generate a correctionto the SU(Na) gauge kinetic function The instanton amplitude takes on thefollowing form

〈Fa(p1)Fa(p2)〉E2 =

intd4x d2θ d2micro exp(minusSE2) exp (Z prime

0(E2)) AF 2a(E2 D6a)

where AF 2a(E2 D6a) is the annulus diagram in figure 4 which absorbs all the

appearing fermionic zero modes and where the gauge boson vertex operators inthe (0)-ghost picture have the usual form

V(0)A (z) = ǫmicro (partmicroX(z) + i(p middot ψ)ψmicro(z)) e

ipmiddotX(z) (620)

4By duality this distinction is related to the two kinds of deformations of genus g curvesstudied in [42] The first kind are the curves moving in families ie transversal deformationof the curve The second kind is related to the deformations coming with the genus g of thecurve

21

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

E2D6a

θ(12)

θ(12)

micro(minus12)

microaF

aF

x

x x

x

x

x

(0)

(0) (minus12)

Figure 4 Annulus diagram for E2-instanton correction to fa The upper indicesgive the ghost number of the vertex operators

Analogous to world-sheet instantons these diagrams can be generalised tomulti tr(W 2)h couplings Just from the zero mode counting one immediately seesthat they can be generated by E2-instantons with h sets of complex deformationzero modes of the second kind and no other additional zero modes Then be-sides the annulus diagram in figure 4 there are h minus 1 similar diagrams On theD6a brane one inserts two gauginos in the (+12) ghost picture and on the E2boundary two micro modulinos in the (minus12) ghost picture Clearly once the internalN = 2 superconformal field theory is known as for toroidal orbifolds or Gepnermodels these annulus diagrams can be computed explicitly They involve up tofour-point functions of vertex operators on an annulus world-sheet with the twoboundaries on the E2 and the D6a brane Very similar to the N = 2 open stringsectors for loop-corrections to fa one expects these instanton diagrams to alsocontain a sum over world-sheet instantons Therefore the generic E2-instantoncontribution to the holomorphic gauge kinetic functions has the moduli depen-dence fnp

(eminusUc

I eminusT ci

)

64 Instanton corrections to the FI-terms

Having shown that E2-instanton corrections to the gauge couplings are possibleit is natural to investigate whether such instantons also contribute to the FI-termsfor the U(1) gauge symmetries on the D6-branes As we have seen in section 4 atthe one-loop level the contributions to the gauge couplings and to the FI-termshave the same functional form

Assume now that as in the last section in the background with intersectingD6-branes we can find an E2-instanton with only two θ fermionic zero modes andtwo additional fermionic zero modes related to a deformation of the E2 If nowsimilar to the D6-branes we could break supersymmetry on the E2-branes by aslight deformation of the complex structure then we would expect four θ-likefour micro-like and two y-like zero modes As shown in figure 5 these could generatean FI-term on the D6-branes However since the E2-brane must be invariantie an O(1) instanton under the orientifold projection a complex structuredeformation does not necessarily break supersymmetry on the E2-instanton In

22

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

this case the analogous situation to the one-loop D6-brane generation of theFI-term cannot happen

However there is another mechanism to generate an FI-term on the E2-instanton namely by turning on the

intΞC3 modulus through the three-cycle the

E2-instanton is wrapping This also appears in the (generalised) calibration con-dition [43 44] for supersymmetry on the E2-brane Therefore it is possible thatthe one-loop diagram in figure 5 indeed generates an FI-term on the D6a braneonce the C3 flux through the E2 is non-zero

D6a E2

x

x

θ(12)

θ(12)

(12)

(12)

micromicro

(minus12)

(minus12)

(minus12)

(minus12)

x

x

x

x

x

x

x

aDθ

θ micro

micro(0)

Figure 5 Annulus diagram for E2-instanton correction to ξa The upper indicesgive the ghost number of the vertex operators

Here we will leave a further study of the concrete instanton amplitudes for gminus2a

and ξa and their relation for future work and conclude that just from fermioniczero mode counting we have evidence that E2-instanton corrections to both thegauge kinetic functions and the FI-terms are likely to appear

7 Conclusions

In this paper we have investigated a number of aspects related to loop and D-brane instanton corrections to intersecting D6-brane models in Type IIA orien-tifolds In particular we have revisited the computation of one-loop correctionsto the FI-terms

Using results for the gauge threshold corrections in intersecting D6-branemodels on a toroidal orientifold we explicitly computed the Wilsonian holomor-phic gauge coupling in this setup On the way exploiting holomorphy and theShifman-Vainshtein respectively Kaplunovsky-Louis formula it was possible toconstrain the form of the matter field Kahler metrics In the second part wediscussed E2-brane instanton corrections to the superpotential the gauge kineticfunction and the FI-terms For the first we showed in which sense one can extractthe form of the holomorphic superpotential from a superconformal field theorycorrelation function of matter fields in the E2-instanton background

Moreover we showed that E2-instantons wrapping a three-cycle which hasprecisely one complex deformation and no matter zero modes can in principle

23

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

contribute to the gauge kinetic function for a gauge theory on a stack of D6-branes By turning on the R-R three-form modulus also instanton correctionsto the FI-terms become possible A more detailed investigation of the appearingannulus diagrams is necessary to eventually establish the appearance of theseinstanton corrections but our first steps indicate that such corrections are indeedpresent in N = 1 D-brane vacua

Acknowledgements

We would like to thank Emilian Dudas Michael Haack Sebastian MosterErik Plauschinn Stephan Stieberger Angel Uranga and Timo Weigand for in-teresting discussions This work is supported in part by the European Com-munityrsquos Human Potential Programme under contract MRTN-CT-2004-005104lsquoConstituents fundamental forces and symmetries of the universersquo

24

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[3] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[4] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalstring compactifications with D-branes orientifolds and fluxesrdquohep-th0610327

[5] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo hep-th0302221

[6] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for Intersecting D-branes RevisitedrdquoarXiv07052150 [hep-th]

[7] A Lawrence and J McGreevy ldquoD-terms and D-strings in open stringmodelsrdquo JHEP 10 (2004) 056 hep-th0409284

[8] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[9] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[10] G Lopes Cardoso and B A Ovrut ldquoA Green-Schwarz mechanism for D =4 N=1 supergravity anomaliesrdquo Nucl Phys B369 (1992) 351ndash372

[11] L E Ibanez and D Lust ldquoDuality anomaly cancellation minimal stringunification and the effective low-energy Lagrangian of 4-D stringsrdquo Nucl

Phys B382 (1992) 305ndash364 hep-th9202046

[12] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

25

[13] V Kaplunovsky and J Louis ldquoOn Gauge couplings in string theoryrdquoNucl Phys B444 (1995) 191ndash244 hep-th9502077

[14] R Blumenhagen M Cvetic and T Weigand ldquoSpacetime instantoncorrections in 4D string vacua - the seesaw mechanism for D-branemodelsrdquo Nucl Phys B771 (2007) 113ndash142 hep-th0609191

[15] L E Ibanez and A M Uranga ldquoNeutrino Majorana masses from stringtheory instanton effectsrdquo JHEP 03 (2007) 052 hep-th0609213

[16] M Haack D Krefl D Lust A Van Proeyen and M ZagermannldquoGaugino condensates and D-terms from D7-branesrdquo JHEP 01 (2007) 078hep-th0609211

[17] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[18] M Bianchi F Fucito and J F Morales ldquoD-brane Instantons on theT6Z3 orientifoldrdquo arXiv07040784 [hep-th]

[19] M Cvetic R Richter and T Weigand ldquoComputation of D-braneinstanton induced superpotential couplings Majorana masses from stringtheoryrdquo hep-th0703028

[20] L E Ibanez A N Schellekens and A M Uranga ldquoInstanton InducedNeutrino Majorana Masses in CFT Orientifolds with MSSM-like spectrardquoarXiv07041079 [hep-th]

[21] M Billo et al ldquoClassical gauge instantons from open stringsrdquo JHEP 02

(2003) 045 hep-th0211250

[22] B Florea S Kachru J McGreevy and N Saulina ldquoStringy instantonsand quiver gauge theoriesrdquo hep-th0610003

[23] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo hep-th0612132

[24] R Argurio M Bertolini G Ferretti A Lerda and C Petersson ldquoStringyInstantons at Orbifold Singularitiesrdquo arXiv07040262 [hep-th]

[25] M Bianchi and E Kiritsis ldquoNon-perturbative and Flux superpotentials forType I strings on the Z3 orbifoldrdquo hep-th0702015

[26] M Bianchi S Kovacs and G Rossi ldquoInstantons and supersymmetryrdquohep-th0703142

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[27] E Witten ldquoWorld-sheet corrections via D-instantonsrdquo JHEP 02 (2000)030 hep-th9907041

[28] M B Green and M Gutperle ldquoEffects of D-instantonsrdquo Nucl Phys

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[29] M Gutperle ldquoAspects of D-instantonsrdquo hep-th9712156

[30] P Anastasopoulos M Bianchi G Sarkissian and Y S Stanev ldquoOn gaugecouplings and thresholds in type I gepner models and otherwiserdquo JHEP 03

(2007) 059 hep-th0612234

[31] E Poppitz ldquoOn the one loop Fayet-Iliopoulos term in chiral fourdimensional type I orbifoldsrdquo Nucl Phys B542 (1999) 31ndash44hep-th9810010

[32] P Bain and M Berg ldquoEffective action of matter fields in four-dimensionalstring orientifoldsrdquo JHEP 04 (2000) 013 hep-th0003185

[33] M A Shifman and A I Vainshtein ldquoSolution of the anomaly puzzle insusy gauge theories and the Wilson operator expansionrdquo Nucl Phys B277

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[34] D Lust P Mayr R Richter and S Stieberger ldquoScattering of gaugematter and moduli fields from intersecting branesrdquo Nucl Phys B696

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[35] B Kors and P Nath ldquoEffective action and soft supersymmetry breakingfor intersecting D-brane modelsrdquo Nucl Phys B681 (2004) 77ndash119hep-th0309167

[36] I Antoniadis C Bachas and E Dudas ldquoGauge couplings infour-dimensional type I string orbifoldsrdquo Nucl Phys B560 (1999) 93ndash134hep-th9906039

[37] M Berg M Haack and B Kors ldquoLoop corrections to volume moduli andinflation in string theoryrdquo Phys Rev D71 (2005) 026005hep-th0404087

[38] D Cremades L E Ibanez and F Marchesano ldquoYukawa couplings inintersecting D-brane modelsrdquo JHEP 07 (2003) 038 hep-th0302105

[39] M Cvetic and I Papadimitriou ldquoConformal field theory couplings forintersecting D-branes on orientifoldsrdquo Phys Rev D68 (2003) 046001hep-th0303083

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[40] S A Abel and A W Owen ldquoInteractions in intersecting brane modelsrdquoNucl Phys B663 (2003) 197ndash214 hep-th0303124

[41] D Cremades L E Ibanez and F Marchesano ldquoComputing Yukawacouplings from magnetized extra dimensionsrdquo JHEP 05 (2004) 079hep-th0404229

[42] C Beasley and E Witten ldquoNew instanton effects in string theoryrdquo JHEP

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[43] J Gutowski G Papadopoulos and P K Townsend ldquoSupersymmetry andgeneralized calibrationsrdquo Phys Rev D60 (1999) 106006 hep-th9905156

[44] F Gmeiner and F Witt ldquoCalibrated cycles and T-dualityrdquomathdg0605710

28