저 시-비 리- 경 지 2.0 한민

는 아래 조건 르는 경 에 한하여 게

l 저 물 복제, 포, 전송, 전시, 공연 송할 수 습니다.

다 과 같 조건 라야 합니다:

l 하는, 저 물 나 포 경 , 저 물에 적 된 허락조건 명확하게 나타내어야 합니다.

l 저 터 허가를 면 러한 조건들 적 되지 않습니다.

저 에 른 리는 내 에 하여 향 지 않습니다.

것 허락규약(Legal Code) 해하 쉽게 약한 것 니다.

Disclaimer

저 시. 하는 원저 를 시하여야 합니다.

비 리. 하는 저 물 리 목적 할 수 없습니다.

경 지. 하는 저 물 개 , 형 또는 가공할 수 없습니다.

이학박사학위논문

Non-perturbative Dualities

in Various Dimensions

여러가지 차원에서의 비섭동적인 양면성

2015년 8월

서울대학교 대학원

물리천문학부 물리학전공

김 정 민

Abstract

Non-perturbative Dualities

in Various Dimensions

Jungmin Kim

Department of Physics & Astronomy

The Graduate School

Seoul National University

The thesis is to study examples in which strongly-coupled phenomena and their

non-perturbative dualities in various dimensions can be studied in detail by using

supersymmetry.

Firstly, I study ‘6d little string theories’ originated from the fundamental

strings near type II NS5-branes in the decoupling limit of gravitational interac-

tion. I studied 2d gauge theory descriptions of type II little strings. For IIA little

strings, I investigate new 2d N = (0, 4) gauge theories more relevant to study IR

physical observables. Using these 2d gauge theories, I exactly compute the ellip-

tic genus of IIA / IIB little string theories compactified on a circle, and confirm

T-duality exchanging winding strings and the Kaluza-Klein momentum.

I also study vortices in Higgs phases of 3d N = 4 and 3 supersymmetric gauge

theories, and exactly compute their SUSY partition functions. They are related

to the relevant deformations of IR superconformal fixed points. N = 4 partition

function confirms the proposed Seiberg dualities and suggests non-trivial exten-

sions. In N = 3 theories with Chern-Simons interactions, I verify some properties

of non-topological vortices in some simple cases.

Keywords : Duality, Supersymmetry, String theory, Little string, Vortex, Soliton

Student Number : 2010-23143

i

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Dualities in String theory . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Superstring theories . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 Type II string theories and T-duality . . . . . . . . . . . . . . 5

1.1.3 SL(2,Z) duality in type IIB string theory . . . . . . . . . . . 8

1.1.4 M-theory and String duality . . . . . . . . . . . . . . . . . . . 11

1.2 Electromagnetic duality in 4 dimensions . . . . . . . . . . . . . . . . 22

1.2.1 Electromagnetic duality . . . . . . . . . . . . . . . . . . . . . 22

1.2.2 SL(2,Z) duality in 4d N = 4 Supersymmetric gauge theories 27

1.2.3 Seiberg duality in 4d N = 1 Super QCDs . . . . . . . . . . . 32

II. Little strings and T-duality . . . . . . . . . . . . . . . . . . . . . . . 36

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.1.1 DLCQ description of type II little string theories . . . . . . . 38

2.2 IIB little strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.2.1 A brief review . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.2.2 The elliptic genus of IIB little strings . . . . . . . . . . . . . . 47

2.3 IIA little strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.3.1 N = (0, 4) gauge theory descriptions . . . . . . . . . . . . . . 53

2.3.2 The elliptic genus of IIA little strings . . . . . . . . . . . . . 58

2.4 T-duality of protected little string spectra . . . . . . . . . . . . . . . 60

2.4.1 One NS5-brane . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.4.2 Two NS5-branes . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.4.3 Three NS5-branes . . . . . . . . . . . . . . . . . . . . . . . . 76

2.5 SL(2, Z) transformations of the elliptic genus . . . . . . . . . . . . . 82

ii

2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

III. 3d Seiberg duality and Vortices . . . . . . . . . . . . . . . . . . . . 86

3.1 Vortices in 3d gauge theories . . . . . . . . . . . . . . . . . . . . . . 86

3.2 Supersymmetric gauge theories and vortices . . . . . . . . . . . . . . 87

3.3 Vortex partition functions of 3d gauge theories . . . . . . . . . . . . 93

3.3.1 Supersymmetric gauge theories and vortices . . . . . . . . . . 93

3.3.2 Vortex quantum mechanics . . . . . . . . . . . . . . . . . . . 105

3.3.3 N = 4 and 3 indices for vortices . . . . . . . . . . . . . . . . 109

3.4 Seiberg dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

3.4.1 N = 4 dualities from vortices . . . . . . . . . . . . . . . . . . 118

3.4.2 Aspects of N = 3 dualities from vortices . . . . . . . . . . . . 127

3.5 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .136

I. N = (4, 4) gauge theory of IIB strings . . . . . . . . . . . . . . . . .137

II. SUSY and cohomological formulation for vortex quantum me-

chanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .139

B.1 Saddle points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

B.2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .152

초록 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .160

iii

Chapter 1

Introduction

Studies on quantum field theories traditionally have depended on the perturba-

tion theory. In spite of progress on various perturbative approaches, there are still

interesting strongly-coupled phenomena which can not be studied with the pertur-

bation theory. We can find examples of various phase transitions in strongly-coupled

gauge theories. One of the important examples is ‘quark confinement phenomenon’

in QCD. Though non-perturbative studies can help to understand various strongly-

coupled phenomena, the systematic methods of them have not yet fully developed.

In this thesis, I study examples of the supersymmetric systems. Supersymmet-

ric field theories realize various interesting non-perturbative phenomena analogous

to them in nature. However, supersymmetric systems have good properties which

make possible non-perturbative studies. Firstly, quantum effects are well-controlled

even in the strongly-coupled regime, we can more easily analyze non-perturbative

phenomena. Recently, it has been established to systematically compute physical

quantities protected from quantum corrections in supersymmetric systems. It is

called ‘localization method’. Besides on this technical advantage, supersymmetric

systems sometimes provide the other advantage to non-perturbative studies. It is

the use of ‘duality’.

Duality in physics means existence of diverse formulations on a physical sys-

tem. We call these different theories as ‘dual theories’. Dualities can be often very

useful in case that different ‘dual’ theories provide easier and more appropriate de-

scriptions to strongly-coupled systems. Moreover, dualities provides very powerful

tools to compute characteristic physical quantities related to these phenomena. For

example, the critical temperature in 2d Ising model can be easily obtained from

the self-dual temperature between two dual theories, and the result perfectly agree

1

to Onsagar’s exact solution.

In case that a physical system has weak-strong coupling dualities, it can be

a great advantage to describe the non-perturbative phenomena by using the dual

weakly-coupled perturbative theories. Generally, clarifying existence of such a du-

ality is very hard and need concrete non-perturbative studies. However, in case of

supersymmetric systems, various weak-strong coupling dualities has been proposed

and studied by using supersymmetry. An example is the exact SL(2,Z) electromag-

netic duality in 4d N = 4 supersymmetric gauge theory. I will review this duality

in detail in this chapter.

Electromagnetric duality as a weak-strong coupling duality. Electromagnetric

duality is realized as a complete form in 4d N = 4 supersymmetric Yang-Mills the-

ories, and forms SL(2,Z) duality group in which W-bosons, magnetic monopoles,

and various dyons are mutually exchanged. In various 4d N = 2 supersymmet-

ric gauge theories, electromagnetic duality has been extensively studied [1]. For

4d N = 1 supersymmetric gauge theories, there is ‘Seiberg duality’ in which two

different gauge theories describe the same IR physics. Seiberg duality is also a

weak-strong coupling duality very analogous to electromagnetic duality. Besides of

these dualities, more various weak-strong coupling dualities in the supersymmetric

systems have been proposed and supported with detailed non-perturbative results.

Moreover, as the non-perturbative aspects of supersymmetric field theories can

be studied in detail, various strong-strong coupling dualities have been studied. Of

course, strong-strong coupling dualities can not have any aspects that we can study

perturbatively. Therefore, the systematic non-perturbative method are seriously

needed to study the strong-strong coupling dualities. In this thesis, I study ‘6d

little string theories’ and their T-duality, dynamics of topological vortices near the

IR strongly-coupled fixed point of 3d N = 4 supersymmetric gauge theories and

their 3d ‘Seiberg duality’. All of the physical systems I deal with in this thesis have

the ‘strong-strong coupling dualities’.

For many years, dualities in supersymmetric systems were studied and discov-

ered from the perspective of superstring theories. In this case, dualities are directly

2

related to non-perturbative dualities of string theories and their dynamics. Duali-

ties in superstring theories have been also revealed by the non-perturbative studies

related with ‘D-branes’. These studies have extended and deepened our under-

standing on string theories. Non-perturbative dualities in string theories will also

reviewed in this chapter.

Various supersymmetric systems can be engineered on intersection of the brane

systems in string theories. Physical systems which I study in this thesis are also

engineered by degrees of freedom localized on the brane-systems in the string the-

ories. In these cases, dualities between the engineered supersymmetric systems are

inherited from the bulk string theory, or realized as accidental dualities due to re-

arrangement of branes. In these situations, viewpoints of string theories are very

useful to understand ‘strong-strong coupling dualities’ between these supersym-

metric systems. For this studies, I will also elaborately use supersymmetry.

The thesis is organized as follows. In the chapter 2, I introduce ‘6d little string

theories’ as UV-completion of the quantum field theories on NS5-branes in string

theories. Their existence is predicted from the fundamental strings bounded on

NS5-branes. For a definite examples, I consider ‘Type IIB little string theories’ and

the non-perturbative solitonic instanton strings known as ‘IIB little strings’. I also

introduce 6d ‘IIA little string theories’ from IIA NS5-branes. I studied the non-

perturbative BPS partition function of the little strings on a compactified spatial

circle, and T-duality between the IIA and IIB little string theories in which the

momenta on the circle and the winding number of strings are exchanged each other.

In the chapter 3, I introduce the 3 dimensional gauge theories and solitonic

vortex particles which govern strongly-coupled IR regime of the theories. For a

concrete example, I study 3d N = 4 and 3 supersymmetric gauge theories and their

vortices, and brane constuction of the theories in IIB string theory. I introduce 3d

Seiberg dualities in which two different supersymmetric gauge theories describes

the same strongly-coupled physics in IR conformal fixed point. Using the effective

quantum mechanics model of the vortices, I study the BPS partition function of

the vortices, and clarify the role of the vortices in Seiberg dualities.

3

1.1 Dualities in String theory

1.1.1 Superstring theories

1There exist five kinds of superstring theories, and their perturbative descriptions

are so different each other. There are two kind of type II string theories, type

IIA and IIB string theory, and there are two kind of heterotic string theories with

SO(32) or E8×E8 gauge symmetry. These type II and heterotic string theories deal

with dynamics of the oriented strings. For dynamics of the unorientied strings, there

is type I string theory. These superstring theories can be classified by 2d conformal

field theory descriptions on strings’ worldsheet.

Firstly, 2d worldsheet conformal field theory of type II strings is given by 2d

N = (1, 1) superconformal field theory, called as ‘Ramond-Neveu-Schwarz(RNS)’

string. By GSO projection, one can consistently remove the tachyonic ground state,

and guarantee the stable vacuum and spacetime supersymmetry. In differing GSO

projection, one can obtain type IIA or IIB superstring theory, with 10 dimensional

N = (1, 1) or N = (2, 0) spacetime supersymmetry, respectively.

On the other hand, since the left movers and the right movers on string’s

worldsheet decouple to each other, different field contents can be introduced for

each worldsheet chirality. Heterotic strings has the worldsheet with different field

contents in regard of the worldsheet chirality. The left moving sector has 26 bosonic

fields, while the right moving sector has the same field contents of the type II string.

To give spacetime interpretation for 10 bosonic fields in each worldsheet chiral sec-

tor, the remaining 16 bosonic fields in the left moving sector should be compactified

on 16-torus with the even self-dual lattice. For this reason, inevitablely, heterotic

string theories involves 10 dimensional gauge supermultiplet with SO(32) or E8×E8

gauge symmetry.

Type II strings are oriented strings. From the orientifold projection of type

II strings, one can obtain another string theory, which involve unoriented strings.

1In this chapter, I abopted notation in [2].

4

This string theory is called as ‘type I string theory’. One of the main achievement

in string theory is that the gravitational anomaly in type I string theory can be

cancelled by introducing 10 dimensional gauge multiplet with SO(32) gauge group,

and type I string theory can be a consistent string theory. The microscopic origin

of the 10d gauge multiplet can be explained, as follows.

Unoriented strings in type II string theory can be obtained by introducing

a spacetime filling orientifold plane(O9-plane). However, this plane has −16 D9-

brane charge, and can be a source of gravitational anomaly. By introducing more 16

spacetime-filling D9-branes, the gravitional anomaly can be cancelled, and obtain

the consistent vacuum of the unoriented strings. D9-O9 backgroud should carry 10

dimensional gauge multiplet with SO(32) gauge group. Since D9-branes are 12 -BPS

and stable only in type IIB string theory, type I string theory can be obtained from

orientifold projection of type IIB string theory.

1.1.2 Type II string theories and T-duality

For type II closed strings, one can impose periodic(or R-sector) or anti-periodic

boundary condition(or NS-sector) to the worldsheet fermions. For two different

chirality, NS-NS sector in massless spectrum give us the following gravity fields,

gµν , Bµν , Φ (1.1)

Each field is called as the graviton, the Kalb-Ramond 2-form gauge fields, and

the dilaton field, respectively. Additionally, RR-sector in massless spectrum give

us various form fields, Cn, with gauge symmetry. With the massless fermions in

10d from R-NS sector and NS-R sector, These massless fields consists of type II

supergravity.

Each form fields couple to various fundamental objects in string theory, as the

following form of Maxwell equations in 10 dimensions,

dF = ⋆Jm , d ⋆ F = ⋆Je (1.2)

5

where F = dA, and Jm,e are magnetic or electric currents, respectively. Electric

charges and magnetric charges are constrained by the following Dirac quantization

condition,

µe · µm ∈ Z . (1.3)

The fundamental strings electrically couple to the Kalb-Ramond 2-form field B2,

and non-perturbative magnetic 5 dimensional objects, called by NS5-branes, couple

to B2 as magnetic objects. For RR form fields, various D-branes couples to them

electrically or magnetically. Dp-branes are also non-perturbative objects with p

dimensionality in string theories. For n-form RR fields, D(n−1)-branes can couple

as electric objects, and D(7− n)-branes can coupled as magnetic objects.

Type IIA supergravity has RR 1-form C1, and 3-form field C2. Type IIB super-

gravity has 0-form scalar field C0, 2-form RR field C2, and 4-form RR field C4 with

self-dual 5-form field strength. By definition, tension of the fundamental string is

given by,

TF1 =1

2π(ℓs)2(1.4)

In addition, one can obtain the following tensions of D-branes and NS5-branes,

TDp =1

(2π)p(ℓs)p+1gs, TNS5 =

1

(2π)5(ℓs)6g2s(1.5)

where ℓs is the string length scale, and gs is the string coupling constant. The above

tension formula can be obtained by several different viewpoints. Firstly, branes

are sources of various form fields, and identified as blackbranes in supergravity

background. The brane tension can be obtained by the Gauss’ low in the blackbrane

backgrounds. In viewpoint of 10 d spacetime supersymmetry, these branes are

so-called half-BPS objects, which preserve 1/2 supersymmetry. From the central

charges of supersymmetry algebra, brane tensions can be identified, and they are

exact even at quantum level.

In other viewpoint, D-branes can interact each other with exchanging strings.

By computing string amplitude with cylindrical worldsheet between two parallel

6

identical brane, D-brane tensions are identified.

It can be regarded that branes give non-perturbative corrections to string

scattering amplitudes. In string perturbation theory, quantum effects are con-

trolled by order of the string coupling constant gs. Notice that NS5-branes give

the usual non-perturbative correction of order ∼ e− 1

g2s , and D-branes give another

non-perturbative corrections as ∼ e− 1

gs . Therefore, studying dynamics of branes is

essential to understand non-perturbative aspects of string theories.

From now, I demonstrate non-perturbative dualities in string theories. The

bosonic action of IIA supergravity is given by

SNS =1

2κ2

∫d10x

√−ge−2Φ

(R+ 4(∇Φ)2 − 1

2|H3|2)

)(1.6)

SR,IIA = − 1

2κ2

∫d10x

√−g(12|F2|2 +

1

2|F4|2

)− 1

4κ2

∫B2 ∧ F4 ∧ F4 (1.7)

Type IIB supergravity has the same action in NS-NS sector. In RR sector, the

action is given by,

SR,IIB = − 1

2κ2

∫d10x

√−g(12|F1|2+

1

2|F3|2+

1

4|F5|2

)− 1

4κ2

∫C4∧H3∧F3 (1.8)

2 where H3 = dB2, Fn = dCn−1. Note that

F4 = dC3 + C1 ∧H3 (1.9)

F3 = F3 − C0H3 , F5 = F5 −1

2C2 ∧H3 +

1

2B2 ∧ F3 (1.10)

Note that the string coupling constant is dynamically given by gs =⟨eΦ⟩. The

Newton constant is given by,

16πG10 = 2κ210 =1

2π(2πℓs)

8g2s ≡ 2κ2gs (1.11)

2The self-duality condition for F5 = ⋆F5 should be given as an extra contidion. The action givethe correct equation of motion for type IIB supergravity

7

If taking compactification along a spatial circle in type II string theories, and impose

the following relation of the radii of spatial circles in each type II string theories,

RIIB ·RIIA = α′ = (ℓs)2, (1.12)

it has been known that one can obtain the unique 9 dimensional string theory. It

is called ‘T-duality’. In this duality, the Kaluza-Klein momentum on the circle and

the winding strings are exchanged each other, as follows,

mwinding = 2πRIIA,B · TF1 =RIIA,B

ℓ2s=

1

RIIB,A= mKK−momentum . (1.13)

This is a kind of perturbative duality, however, this duality has been regarded as the

exact non-perturbative duality between string theories, with supported by various

non-trivial non-perturbative tests. In T-duality, the string coupling constants relate

to each other by the following relation,

gIIB,A =ℓs

RIIA,BgIIA,B , (1.14)

respectively. 3

One of the main purpose of the thesis is to show T-duality where interaction

of the fundamental strings has an extreme strong coupling. Since the NS5-branes

can be a source of the dilaton field, and the dilaton field is divergent near NS5-

branes, the fundamental strings near NS5-branes are very strongly coupled. These

fundamental strings moving on NS5-branes are called ‘little string’. I will study

T-duality of type II little strings in detail in the chapter 2.

1.1.3 SL(2,Z) duality in type IIB string theory

Besides T-duality, it has been known that type IIB string theory has SL(2,Z)

duality symmetry. This plays a role of weak-strong coupling duality in type IIB

3This relation can be obtained from the NS-sector of the type II supergravity action.

8

string theories. SL(2,Z) duality has been regarded as the exact duality in type IIB

string theory, with supported by various non-perturbative tests.

Type IIB supergravity has SL(2,R) global symmetry, as a classical symmetry.

By Dirac quantization condition, SL(2,Z) is remained as the true quantum duality

symmetry. Type IIB supergravity fields are in SL(2,Z) representation. The Kalb-

Ramond 2-form field B2, and 2-form field in RR-sector C2, consist of a doublet in

SL(2,Z) transformation, as follows,

B2 =

B(1)2

B(2)2

(1.15)

where B(1)2 = B2 and B

(2)2 = C2. The self-dual RR 4-form field is the singlet in

SL(2,Z). The RR 0-form field and the dilaton field form the complex axio-dilaton

field, as follows,

τ = C0 + ie−Φ , (1.16)

If representing an element of SL(2,Z) as

Λ =

d c

b a

∈ SL(2,Z) , (1.17)

the axio-dilaton field transforms, as follows,

τ −→ aτ + b

cτ + d. (1.18)

Defining the following matrix form of the axio dilaton field,

M = eΦ

|τ |2 −C0

−C0 1

(1.19)

the axio-dilaton field transforms as an symmetric representation under SL(2,Z),

9

as follows,

M −→ (Λ−1)MΛ−1 (1.20)

In these notation, one can obtain manifestly SL(2,Z) invariant action of type IIB

supergravity, as follows,

S =1

2κ2

∫d10x

√−g(R− 1

12HTµνρMHµνρ +

1

4tr(∂µM∂µM−1)

)− 1

8κ2

(∫d10x

√−g|F5|2 +

∫ϵijC4 ∧H(i)

3 ∧H(j)3

)(1.21)

In type IIB superstring theory, the fundamental string and D1-brane charges can

be represented as SL(2,Z) vectors,

µF1 = (1, 0) , µD1 = (0, 1) . (1.22)

SL(2,Z) duality symmetry imply that existence of generalized (p, q)-strings which

has p fundamental string charges and q D1-brane charges, where p and q are rela-

tively prime, the charges as SL(2,Z) vectors and string tension are given by,

µ(p,q)−string = (p, q) , T(p,q)−string = TF1|p− qτB| (1.23)

where τB is the complexified string coupling constant in type IIB string theory,

given by

τB = ⟨τ⟩ = θ02π

+i

gs(1.24)

Type IIB fivebranes also has NS5-brane charges and D5-brane charges, simulta-

neously. These are called (p, q)-fivebranes, which has SL(2,Z) charge vectors and

brane tension.

µ(p,q)−fivebranes = (p, q) , T(p,q)−fivebranes = TNS5|p− qτB| (1.25)

Besides 12 -BPS (p, q)-strings and fivebranes, there exist 1

4 -BPS string(or fivebrane)

junctions, 12 -BPS (p, q)-sevenbranes, which are sources of the axio-dilaton fields,

10

and etc. Existence of (p, q)-sevenbranes implies that type IIB string theory can be

understood as another 12 dimensional theory compactified on a torus fiberation,

parametrized by the value of the axio-dilaton field. The 12 dimensional theory is

called ‘F-theory’. ‘F-theory’ provides the UV origin of various (p, q)-sevenbranes.

(p, q)-sevenbranes are geometrized in F-theory as degeneration points in transverse

R2 with the torus fiberation, which corresponds to the branch points of the axio-

dilaton field on R2 in type IIB string theory. It has been accepted that all the

spectrum in type IIB branes form SL(2,Z) multiplets, and type IIB string theory

has the exact SL(2, Z) duality symmetry.

As a specific element of SL(2,Z),

S =

0 1

−1 0

, (1.26)

under the above transformation, the string coupling constant transforms

τB −→ − 1

τB(1.27)

This duality is called as ‘S-duality’ [3]. This exchanges the strong coupling regime

and the weak-coupling regime of type IIB string theory. In S-duality, the Ramond

sector and the NS sector exchanges their roles in the dynamics. This suggests that

type IIB string theory is a UV complete string theory, as its own.

1.1.4 M-theory and String duality

M-theory as the strong-coupling type IIA string theory

The five string theories(Type IIA and IIB, type I, and heterotic string theo-

ries) has very different physical aspects, especially between type II and heterotic

string theories. However, it has been proposed that in the strong coupling limit

of type IIA supergravity, additional spatial circle emerge, and type IIA supergrav-

ity become eleven dimensional supergravity on circle [4]. This implies that in the

11

strong coupling limit of type IIA string theory, there can be emerged 11 dimen-

sional consistent quantum theory, which is called as ‘M-theory’. M-theory is not

a string theory any more. M-theory involves quantum membrane spectrum, called

as ‘M2-brane’, and M2-branes on a small M-theory circle can be identified as type

IIA fundamental strings.

The first key clue of M-theory is D0-branes in type IIA string theory [4].

Charges in Ramond sector are involved as the central charges of type II superalge-

bra. For D0-brane charges appears in type IIA superalgebra, as follows,

Qα, Qα = δααZ (1.28)

where α and α are the chiral spinor index and the anti-chiral index in 10 dimensions.

The exact formula of Z can be obtained in several ways. One possible consid-

eration is that the charged extremal black hole always holds GM2 ≥ const · |Z|2.

Since G10 ∼ g2s , one can know that the formula of Z. Alternatively, I will represent

Z from D0-brane mass formula of (1.5),

Z =n

gsℓs, n ∈ Z . (1.29)

In [4], E. Witten interprete this infinite tower of D0-branes’ spectrum as the Kaluza-

Klein momenta of the 11 dimensional supergravity on a compactified circle, with a

radius of the circle

RM = gsℓs . (1.30)

In the strong coupling regime in type IIA supergravity, the M-theory circle become

larger and 11d supergravity description is more vaild.

The above correspondence has been also supported by detailed evidence [4].

Firstly, in consideration of low energy dynamics, it can be checked whether Kaluze-

Klein reduced 11d supergravity corresponds to type IIA supergravity. 11d super-

12

gravity action is give by

2κ211S =

∫d11x

√−g11

(R− 1

2|F4|2

)− 1

6

∫A3 ∧ F4 ∧ F4 , (1.31)

where A3 is the 3-form gauge field, F4 = dA3 is the field strength. If I introduce

following gravity field,

ds2 = e−23Φg(10)µν dxµdxν + e

43Φ(dx11 + Cµdx

µ)2 . (1.32)

The following relation can be obtained,

RM = g2/3s ℓp = gsℓs (1.33)

where ℓp is the Planck length scale of 11d supergravity. In viewpoint of type IIA su-

pergravity, Φ is the dilaton field and Cµ is Ramond 1-form field, which corresponds

to the Kaluza-Klein 1-form field of 11d supergravity. Identifying

A(11)µνρ ≡ Cµνρ , A

(11)µν11 ≡ Bµν . (1.34)

I can completely obtain type IIA supergravity (1.6) and (1.7) from 11d supergravity.

Even in a high energy limit, detail evidence of the correspondence for BPS

soliton spectrum are provided in [4]. Existence of the 3-form field A3 in 11d su-

pergravity suggests that there exist two kind of the extremal blackbranes charged

by A3. Electrically charged membrane state is called ‘M2-brane’, and magneti-

cally charged fivebrane state is called ‘M5-brane’. In type IIA supergravity as the

Kaluza-Klein reduced 11d supergravity, D2-branes and NS5-branes correspond to

M2-branes and M5-branes, respectively. The brane tensions are matched each other,

TM2 =1

4π2ℓ3p=

1

4π2ℓ3sgs= TD2 , (1.35)

TM5 =1

(2π)5ℓ6p=

1

(2π)5ℓ6sg2= TNS5 . (1.36)

13

The type IIA fundamental strings are identified by wrapped M2-branes on M-theory

circle,

TF1 =2πRM4π2ℓ3p

=1

2πℓ2s. (1.37)

D6-branes magnetically couple to C1. If one identify C1 as a Kaluza-Klein gauge

field in 11d supergravity, magnetic D6-branes should be geometrized in the trans-

verse 4 directions as the Taub-Nut geometry, given by

ds2 = V (r)(dr2 + r2dΩ22) +

1

V (r)(dy +RM sin2 (θ/2) dϕ)2 , (1.38)

where

V (r) = 1 +RM2r

. (1.39)

This is a 4 dimensional monopole moduli space, realizing M-theory circle as the

circle fiberation on transverse R3. The tension of D6-brane is obtained from the

magnetic flux from Taub-Nut center, which is given by [2]

TD6 =2πRM16πG11

∫d3x∇2V (r) =

(2πRM )2

16πG11=

2π

(2πℓs)7gs(1.40)

The above evidence strongly suggests existence of 11d consistent quantum theory,

as the origin of the high energy limit of 11d supergravity and the strongly coupled

type IIA string theory. This theory is called as ‘M-theory’ [5].

M-theory and U-duality

One of the most important evidence for existence of M-theory is ‘U-duality’.

Taking toroidal compactification from 10 dimensions to d dimensions of type II

supergravity, there are SO(10− d, 10− d;Z) T-duality group. Since this T-duality

can not commute SL(2,Z) duality in type IIB supergravity, T-duality group and

SL(2,Z) group forms a more larger duality group. This extended duality is called

‘U-duality’. The essential statement in [4] is that if regarding ‘U-duality’ as the

fundamental duality group not only in type II supergravity but also in type II

14

string theory, this implies that there exists ‘M-theory’ as 11d consistent quantum

theory, which has 11d supergravity in the low energy limit.

In a toroidal compactification from 10d to 7d of type II supergravity, T-duality

group is SO(3, 3), or equivalently SL(4). U-duality is given by G = SL(5) with

a maximal compact R-symmetry group K = SO(5). The string theory vacuum is

given by G/K, which is the moduli space of type II string theory. In [4], E. Witten

analyzed the central charges in supersymmetry in the infinite limits of subgroups

of SL(5) U-duality. In this case, there exist 10 U(1) gauge fields as a rank 2 anti-

symmetric tensor in SL(5) U-duality. Six U(1) gauge fields come from NS-NS sector

as a rank 2 antisymmetric tensor under SL(4), and four U(1) gauge from RR-sector

as 4 representation in SL(4).

In infinite limits, various combinations of the central charges can be massless,

one can regard some of these phenomena as a partial decompactification. With

various infinite limits of U-duality, one can obtain that various decompactification

limits of type IIA or IIB string theory. For a one-parameter subgroup of SL(5)

U-duality given by,

gt = diag(ec1t, ec2t, ec3t, ec4t, ec5t) , (1.41)

the values of the central charges given by Z(ψij) ∼ e−(ci+cj)t for 1 ≤ i < j ≤ 5.

For a one parameter subgroup given by

gt = diag(e−t, e−t, e−t, e−t, e4t) , (1.42)

in the infinite limit, the six central charges, Z(ψ1,j) with 0 ≤ i < j ≤ 4, vanish.

These become massless spectrum. Moreover, the duality symmetry become SL(4),

which is the correct T-duality group of the 7 dimensional string theory. It is valid

to interpret the emergent massless spectrum in the infinity limit as the elementary

winding strings on a small T 3 in the low energy limit.

If considering

gt = diag(e3t, e3t, e−2t, e−2t, e−2t) , (1.43)

15

In the infinite limit of t, the duality symmetry becomes SL(3)× SL(2), this is U-

duality of the 8 dimensional supergravity. Moreover, the central charges Z(ψ12) →

0 as t → ∞, with which it is appropriate to interpret that this mode represents

the light KK-modes along one of the circles in T 3. This is decompactification of 7

dimensional supergravity to 8 dimensions.

For another example, in the infinity limit of

gt = diag(e2t, e2t, e2t, e−3t, e−3t) , (1.44)

Three central charge of ψij with 1 ≤ i < j ≤ 3 vanish. and duality symmetry group

is SL(2, Z), which is SL(2,Z) in type IIB string theory. It is natural to interpret

three massless spectrum as light Kaluza-Klein modes in the decompactification

limit from 7d to 10d IIB string theory.

All the above examples show that diverse dynamical aspects of type II string

theory are realized in U-duality. Moreover, it was pointed out that a specific infinite

limit in U-duality can be regard as ‘decompactification limit of a four-torus’ in

which the 7 dimensional spacetime is restored to 11d spacetime [4], considering

one-parameter subgroup given by

gt = diag(e4t, e−t, e−t, e−t, e−t) . (1.45)

In t → ∞, four kinds of central charge, Z(ψ1j) with 2 ≤ j ≤ 5, vanish and

SL(4) duality symmetry appears. It is natural that SL(4) is a group of linear

automorphisms of T 4. This has been regard as the first evidence for restoration of

11d Lorentz symmetry from type II string theory.

M-theory and Heterotic string theories

The heterotic supergravity has the graviton field, the Kalb-Ramond 2-form

gauge field, a dilaton field, and SO(32) or E8 × E8 gauge field in the bosonic

16

sector. The heterotic supergravity has the action given by,

S =1

2κ2

∫d10x

√−ge−2Φ

[R+ 4∂µΦ∂

µΦ− 1

2|H3|2 −

κ2

30g2Tradj(|F2|2)

], (1.46)

4 where

H3 = dB2 +ℓ2s4ω3 . (1.47)

ω3 is the Chern-Simons form defined by

ω3 = tr(ω ∧ dω +2

3ω ∧ ω ∧ ω)− 1

30Tradj(A ∧ dA+

2

3A ∧A ∧A) . (1.48)

One can know that

dH3 =ℓ2s4

(trR ∧R− 1

30TradjF ∧ F

). (1.49)

For SO(32) gauge group, there is the 32 dimensional fundamental representation,

and one can obtain that

tr(F ∧ F ) =1

30Tradj(F ∧ F ) . (1.50)

This means that the cohomology class of tr(R∧R) and tr(F∧F ) should be the same,

because the left hand side of the above equation is an exact form. The Yang-Mills

coupling constant is given by

g2YM

4π=g2

4πg2H = (2πℓs)

6g2H (1.51)

where, gH = ⟨eΦ⟩ is the string coupling constant. In [4, 6], it was pointed out that

type II string theories and heterotic string theories can be integrated in M-theory

picture, even though dynamical aspects of type II and heterotic string theories are

very different to each other.

The moduli space of type IIA string theory compactified on K3 surface and

4For anomaly cancellation, the suitable local counter term should be introduced in the action.

17

the heterotic string theory compactified on T 4 are identical, and given by [7]

SO(20, 4;Z)\SO(20, 4;R)/(SO(20)× SO(4)

)(1.52)

This implies that IIA string theory compactified on K3 surface and the heterotic

string theory compactified on T 4 give the same physics. Moreover, in the literature

[4, 6, 5, 8], the detailed non-perturbative tests were provided, and supported the

duality. Especially, it is showed what this type II - heterotic string duality imply

in various situations in string theory, in regard of relation with M-theory [4].

If compactifying the Heterotic string theory on T 2×T 4 to 4 dimensional space-

time, this theory has various 4d N = 4 U(1) vector multiplets5, the antisymmetric

tensor fields, and the graviton field. If decompactifying one of the circles, the het-

erotic string theory compactified on T 5 has various 5d N = 2 U(1) gauge multiplets

which couple to the gravity.

In 5d, there appear more vector multiplets, since the 2-form gauge fields B2 is

dual to A1 due to dB2 = ⋆dA1. In this situation, there appear the non-perturbative

solitonic spectrum carry the self-dual 2-form instanton flux in spatial 4 directions.

These are the instanton solitons as half-BPS states, and their charges are involved

as the central charges in the supersymmetry algebra. These are charged under A1

electrically. The mass spectrum is given by

M ∼ |Z| ∼ |n|g2Hℓs

, n ∈ Z (1.53)

This spectrum comes from the Heterotic NS5-brane compactified on T 4. These

states become very light in the strong coupling limit of the heterotic string theory.

The question in [4] for this spectrum is that in the strong coupling limits, this

states can be interpreted as light Kaluza-Klein modes of a higher dimensional

string theory with 6 dimensions. This question can be answered in string duality.

5The SL(2,Z) exact electromagnetic duality in 4d N = 4 gauge theory can be understoodin viewpoint of the duality of the toroidally compactified heterotic string theory. However, I willdiscuss this duality later, and review in the next chapter.

18

Type IIA string theory compactified on K3×S1IIA is dual to the 5 dimensional

heterotic string theory compactified by T 4 × S1H, the radii of the circles and the

coupling constants are related with each other, by the following relation.

gIIA =1

gH, RIIA =

RH

gH(1.54)

with a fixed moduli of K3 surface. The instanton solitons in the heterotic string

theories on T 4 × S1H corresponds to winding fundamental strings on S1

IIA of type

IIA on K3× S1IIA.

Type IIA string theory compactified on K3 × S1IIA is dual to type IIB com-

pactified on K3× S1IIB by T-duality. Since

gIIB =ℓsRIIA

gIIA , RIIB =ℓ2sRIIA

, (1.55)

From the above relations, it can be obtained that

gIIB =ℓsRH

, RIIB =gHℓ

2s

RH. (1.56)

The instanton solitons of the heterotic on T 5 correspond to the Kaluza-Klein modes

on S1IIB of type IIB string theory compactified on K3× S1

IIB, and these states be-

come lighter in the strong coupling limit of the heterotic string theory. Therefore,

the strong coupling limit gH → ∞ with fixed RH of the 5d heterotic string the-

ory means decompactification of the circle (RIIB → ∞) in type IIB string theory

compactified on K3× S1IIB. This means that the strongly coupled heterotic string

theory with toroidal compactification to 5 dimensions is equivalent to type IIB

string theory compactified to 6 dimensions on K3 surface.

More detailed analysis was provided in [4, 6] directly with type IIB string

theory compactified on K3. Type IIB string theory compactified on K3. Type IIB

on K3 involve five 2-form gauge fields with the self-dual field strengths in R6, and

21 2-form gauge fields with the anti-self-dual field strengths. This has the following

19

moduli space,

N ≡ SO(21, 5;Z)\SO(21, 5;R)/(SO(21)× SO(5)

)(1.57)

This is 105 dimensional moduli space, in which 80 dimensions comes from NS-

sector, 24 dimension comes from R-sector, and 1 dimension from the value of the

dilaton field. Compactifying this on one more circle, the radius of the circle is

involved as a part of the moduli space. Therefore, the moduli space is given by

M = N × R+ (1.58)

where R+ represents the radius of type IIB circle. The T-duality moduli space N

is the same one of the heterotic string theory on R5 × T 5. In case of the heterotic

string theory, R+ comes from the value of string coupling constant. This is regarded

as the first evidence of equvalence between 6d N = (2, 0) gauge theories with

(anti)self-dual 2-form gauge fields and 5d N = 2 gauge theories with the non-

perturbative instanton particles. Near K3 singularities in type IIB string theory,

6d N = (2, 0) superconformal field theories described by the self-dual tensor fields

was found. They have the self-dual tensionless strings as the basic ingredients. In

string duality, it is can be understood that the circle-compactifed 6d N = (2, 0)

theories are equivalent to the non-perturbative 5d gauge theories, constructed by

the dual heterotic string theory compactified on T 5.

One of the first prediction in relation between heterotic string theories and

‘M-theory’ was that the heterotic string theory compactified on T 3 is equivalent to

M-theory compactified on K3 surface. it has been checked that they have exactly

the same moduli space [4], given by

SO(19, 3;R)\SO(19, 3;Z)/(SO(19)× SO(5)

)× R+ (1.59)

In M-theory, R+ means that the radius of K3 surface, but R+ is a moduli coming

from the string coupling constant in the heterotic sting theory. However, I will omit

20

more detailed analysis for this facts.

One of the striking aspects referred in [4, 5] is that the heterotic string theo-

ries and type II string theories have an integrated picture as different geometrical

background in M-theory. In the strong coupling limit of the heterotic string theory

with E8 × E8 gauge symmetry, eleventh direction emerges as a segment with two

boundaries, and E8 gauge symmetries are supported on each boundary, these 10 di-

mensional boundaries are called as ‘Horava-Witten domain wall’ [5]. This segment

with boundaries can be understood as M-theory circle orbifolded by Z2, S1/Z2.

The strongly coupled heterotic string theory on R1,9 with E8 × E8 is equivalent

to M-theory on R1,9 × S1/Z2. The length of the segment is related to physical

parameters in the heterotic string theory, as follows,

RM = gHℓs . (1.60)

On the orbifold sigularities, there should carry 10 dimensional gauge multiplet

with E8 symmetry to cancel the induced gravitational anomaly coming from 11

dimensional bulk supergravity fields. In this viewpoint, the E8×E8 heterotic string

has the UV origin as M2-brane suspend between the Horava-Witten domain walls,

the string tension is given by

TH1 = 2πRMTM2 =1

2π(ℓs)2, (1.61)

Since the heterotic string theories with SO(32) or E8 × E8 gauge symmetries are

T-dual each other6. In addition, [4] has suggested that type I string theory and

SO(32) heterotic string theory are S-dual each other with detailed evidence in the

supergravity and the BPS spectrum. Due to discovery of M-theory as a missing link

between type II string theories and heterotic string theories, all the string theories

are unified as physical aspects of ‘M-theory’.

6T-duality between heterotic string theories holds in the presence of Wilson line expectationvalues turned on a spatial circle, which break SO(32) and E8 ×E8 gauge symmetry to SO(16)×SO(16)

21

1.2 Electromagnetic duality in 4 dimensions

1.2.1 Electromagnetic duality

The Maxwell equations involving the electric current, Jµe , and the magnetic current,

Jµm, are given by2π

g2· ∂µF νµ = Jµe , ∂µ(⋆F )

νµ = Jµm , (1.62)

where (⋆F )µν = 12ϵµνρσF

ρσ and g is the gauge coupling constant. The electromag-

netic duality exchanges the electric fields and the magnetic field as the following

way,

2π

g2· F → 2π

(g′)2· F ′ = −(⋆F ) , (⋆F ) → (⋆F )′ = −2π

g2· F . (1.63)

One can regard this transformation is a weak-strong coupling duality, for the fol-

lowing reason,

g′ =2π

g. (1.64)

The electric current and the magnetic current are also exchanged each other in the

electromagnetic duality,

Je → J ′e = −Jm , Jm → J ′

m = Je . (1.65)

This electromagnetic duality can be generalized as the following fashion.(2π

g2F + i(⋆F )

)→ eiζ

(2π

g2F + i(⋆F )

), (1.66)

(Je + iJm) → eiζ(Je + iJm) . (1.67)

By using the duality transformation, one can set the electric charge and the mag-

netic charge of one fundamental charged particles, as follows,

(µe, µm) = (−1, 0) , (1.68)

22

This particle is called as the electron. This choice is a matter of convention. After

fixing the duality frame ζ, the electric and magnetic charges of the other particles

are determined unambiguously. If all charge spectrum consist of the multiplets

under the electromagnetic duality transformation in a gauge theory, we can choose

physically distinct electromagnetic dual descriptions. In this case, this condition

can be a necessary condition that the theory has the electromagnetic duality. More

strong version of the electromagnetic duality is that all the physical quantities

should form the representations in the electromagnetic duality group.

In a physical system in which the electromagnetic duality exists, the elec-

tromagnetic duality can be extremely useful, since the non-perturbative strongly-

coupled regime of the system can be understood by the weakly-coupled perturbative

description. This duality can be a very special feature link from the weakly-coupled

regime to the strongly-coupled regime of the electromagnetic systems.

Dirac-Zwanziger-Schwinger quantization condition

In the presence of the magnetic monopole and the electric charges, the charges

should be quantized to hold the following condition,

µe · µm ∈ Z (1.69)

One of the easiest way to derive this condition is to use the gauge invariant condition

of the Wilson-loop expectation value near the magnetic monopole, which is defined

as

W = exp

(2πiµe

∮Aµdx

µ

). (1.70)

where the path of the integral is chosen as the equator of a 2-sphere including the

magnetic monopole. By the stokes’ theorem,∮Aµdx

µ =

∫DN

F2 or

∫DS

F2 (1.71)

where DN and DN denote the northern hemisphere and southern hemisphere. one

23

can know that ∫DN

F2 −∫DS

F2 =

∫S2

F2 = µm . (1.72)

The gauge invariant condition of the Wilson loop is that

exp(2πiµe · µm) = 1. (1.73)

Therefore, the Dirac quantization condition is proved. For dyons carrying both a

electric charge and a magnetic charge, one can also drive the similar condition

of the Dirac charge quantization condition. This is called ‘Zwanziger-Schwinger

quantization condition’. This condition is given by

µe1 · µm2 − µe2 · µm1 ∈ Z (1.74)

for two dyons with the charges (µe1, µm1) and (µe2, µm2). These charge quantization

conditions hold in full quantum level.

For a more general consideration, since the magnetic charge violate CP-symmetry,

one can introduce the following generalized Maxwell action with a θ term,

S ∼∫d4x

(− 1

4g2FµνF

µν − θ

32π2ϵµνρσFµνF ρσ

)= − 1

16π

∫d4x Im

(τ(F + i ⋆ F )2

). (1.75)

where the complexifed gauge coupling, τ , is given by,

τ =θ

2π+

2πi

g2(1.76)

This θ term can come from instanton contribution in non-abelian gauge theory

in an abelian Coulomb phase in which the UV non-abelian gauge symmetry is

spontaneously broken to a U(1) abelian gauge symmetries by Higgs mechanism. In

24

this situation, the Maxwell equation is given by

∂µ

(2π

g2F νµ +

θ

2π⋆ F νµ

)= Jνe , ∂µ(⋆F

νµ) = Jνe . (1.77)

θ parametrize distinct vacua of the theory. However, θ → θ + 2π is an exact sym-

metry in the theory. Changing the vacuum adiabatically by growing the value of

θ0, there is an interesting physical phenomenon in which the magnetic monopoles

acquire additional electric charge of ∆θ2π µm, and become the dyons. This is called as

‘Witten effect’ [9]. When θ0 reaches θ0+2π, the system comes back to the original

vacuum, again. Therefore, one can get another electromagnetic duality, called as

‘T’, as follows,

2π

g2F +

θ

2π⋆ F → 2π

g′2F ′ +

θ′

2π⋆ F ′ =

2π

g2F +

(θ

2π+ 1

)⋆ F , (1.78)

⋆ F → ⋆F ′ = ⋆F (1.79)

(µe, µm) → (µe + µm, µm) (1.80)

T-transformation changes τ , as follows,

τ → τ ′ = τ + 1 . (1.81)

The classical electromagnetic duality can be parametrized by ζ parameter in (1.66)

and (1.67). However, in quantum theories, only ∆ζ = π2n (n ∈ Z) transformations

can be compatible with Dirac-Zwanziger-Schwinger quantization condition. When

∆ζ = π2 , this duality transformations means that

2π

g2F +

θ

2π⋆ F → 2π

g′2F ′ +

θ′

2π⋆ F ′ = ⋆F , (1.82)

⋆ F → ⋆F ′ = −(2π

g2F +

(θ

2π+ 1

)⋆ F

)(1.83)

(µe, µm) → (−µm, µe) (1.84)

25

This electromagnetic duality is called as ‘S’-transformation. The gauge coupling

constant transforms under S-transformation,

τ → −1

τ. (1.85)

This is one if the weak-strong coupling duality. These T and S transformations

generate SL(2,Z) electromagnetic duality. Under SL(2,Z), the coupling constant

transforms as

τ → aτ + b

cτ + d(1.86)

where d c

b a

∈ SL(2,Z) (1.87)

The charge spectrum transforms as

(µe, µm) → (aµe + bµm, cµe + dµm) . (1.88)

To take electromagnetic dual descriptions, one should check all the physical observ-

ables of the quantum states form the multiplets under the electromagnetic duality.

Since the electromagnetic duality connects the weak and strong coupling regime, it

is a difficult non-perturbative problem to show the duality. Moreover, in this subsec-

tion, I explained a semi-classical aspects of the electromagnetic duality. We should

consider whether the quantum correction of the system spoil the electromagnetic

duality, or not.

In supersymmetric systems, quantum correction is well-controlled, and there

exists the possibility for the semi-classical electromagnetic duality to be the exact

duality. Expecially, In 4d N = 4 supersymmetric gauge theories, it has been known

that the exact SL(2,Z) electromagnetic duality is realized, and verified by various

non-perturbative tests [3].

26

1.2.2 SL(2,Z) duality in 4d N = 4 Supersymmetric gauge

theories

4d N = 4 supersymmetric gauge theory is described by N = 4 vector supermul-

tiplet, consisting of the gauge field Aµ, 4 chiral fermions λi=1,··· ,4α , and 6 bosonic

real scalars ϕI=1,··· ,6. These theories have SU(4)R ≃ SO(6)R R-symmetry. This is

a global symmetry under which supercharges Qi=1,··· ,4α consist of the fundamental

representation. All field contents are in the adjoint representation of the gauge

group. The action is given by,

S =

∫d4x tr

[− 1

4g2YM

FµνFµν − θ

32π2ϵµνρσF

µνF ρσ − 1

2|Dµϕ

I |2 +g2YM4

[ϕI , ϕJ ]2

+iλαi(σµ)αβDµλ

iβ −

igYM

2(CI)ijλ

αi[ϕI , λjα] +igYM

2(CI)ij λαi[ϕ

I , λαj ]

](1.89)

(CI)ij and (CI)ij are the Clebsch-Gordon coefficients between 4 and 6 representa-

tion in SU(4)R symmetry. N = 4 supersymmetric gauge theory has the maximal

superconformal symmetry, PSU(2, 2|4), and the gauge coupling is exactly marginal.

The low energy physics of the theory is governed by a non-abelian Coulomb phase.

Even in a non-abelian Coulomb phase, the exact SL(2,Z) duality has a role of a

weak-strong coupling duality. SL(2,Z) duality has been verified by non-trivial ev-

idence. Some of the representative studies in SL(2,Z) are in the contexts of string

theory.

For example, 4d N = 4 supersymmetric gauge theory with U(N) gauge group

is realized in the low energy dynamics of D3-branes in type IIB string theory. and

the Yang-Mills coupling constant come from the value of the axio-dilaton field on

D3-branes. In this situation, the electromagnetic duality is realized by the exact

SL(2,Z) duality in type IIB string theory.

Another example is that N = 4 supersymmetric gauge theory is realized by

6d N = (2, 0) superconformal field theory compactified on 2-torus, and SL(2,Z)

duality realized by a geometric duality encoded on 2-torus. Since 6d N = (2, 0)

superconformal field theory is well known as the low energy effective theory of

27

Charge Mass (θ = 0) Mass (θ = 0)

W-boson (µe, 0) |µe · ϕ| |µe · ϕ|Monopole (0, µm)

∣∣∣2πµm·ϕgYM

∣∣∣ |τ µm · ϕ|

dyon (µe, µm)

√(µe · ϕ

)2+(2πµm·ϕgYM

)2|(µe − τ µm) · ϕ|

Table 1: 12 -BPS spectrum of N = 4 supersymmetric gauge theory in the abelian

Coulomb phase.

M5-branes in M-theory. This situation can be more understood in a string theory

context.

Besides in the non-abelian Coulomb phase, SL(2,Z) duality has a manifest

aspect as the electromagnetic duality in the abelian Coulom phase, in which the

gauge group is broken to U(1) gauge groups.

In the abelian Coulomb phase, SL(2,Z) duality has been extensively studied

in the context of heterotic string theory compactified on T 6, or type IIA string

theory compactified on K3 × T 2 has a dynamical sector described by the N = 4

supersymmetric gauge theory [4, 3, 10], and provided various non-perturbative

evidence in BPS spectrum.

SL(2,Z) duality and dyonic spectrum

In 4d U(N) N = 4 supersymmetric gauge theory, the abelian Coulomb phase

is realized by the vacuum expectation value of the scalar fields, ⟨ϕI=1,··· ,6⟩. For an

explicit example, let us assume that ⟨ϕ1⟩ = diag(ϕ1, · · · , ϕN ) ≡ ϕ and ⟨ϕ2,··· ,6⟩ = 0.

In this situation, U(N) gauge symmetry is broken to U(1)N gauge group. The

charge vector of the electric charges and magnetic charges in U(1)N gauge group

can be represented by (µe, µm).

In the abelian Coulomb phase, there can appear the 12 -BPS W-bosons as elec-

trically charged object, and magnetic monopoles and dyons as non-perturbative

spectrum, represented in Table. 1. Since the BPS bound condition implies the fol-

28

lowing triangle inequality for the dyon mass,

M(µe,µm) +M(µ′e,µ′m) ≥M(µe+µ′e,µm+µ′m) . (1.90)

The stable dyons should have the electric charges and the magnetic charges which

are relatively prime.

12 -BPS spectrum forms the mulitplets in SL(2,Z) duality. This called ‘Olive-

Montonen duality’ [11, 12]. At first, this duality had been proposed in a bosonic

field theory. However, there could be things violating this duality. The first one is

that the monopole and W-boson has different spins. Secondly, the mass spectrum

can be spoiled by quantum effects. Moreover, existence of the quantum bound

states for dyons could not be proved. However, in N = 4 supersymmetric gauge

theories, this defects for SL(2,Z) duality can be solved.

Firstly, W-bosons and monopoles are 12 -BPS objects, preserve 1/2 of N = 4

supersymmety. This means that W-bosons and monopoles forms the same massive

supermulitplet in this theory. Therefore, W-bosons and monopoles have the same

spin content in N = 4 supersymmetric gauge theory. Moreover, the gauge coupling

in N = 4 supersymmetric gauge theory is exactly marginal, and the mass spectrum

of BPS spectrum is free from quantum correction by supersymmetry. Therefore,

SL(2, Z) electromagnetic duality can be faithful in N = 4 gauge theory.

Finally, existence of the quantum bound states of dyons is partly proved by

Ashoke Sen [3]. the dyons with 1 magnetic charge and an arbitrary electric charges

are called as ‘Julia-Zee dyon’, and their explicit solution has been already known

[13]. Moreover, Ashoke Sen proved that there exists the unique quantum bound

state of dyons with 2 magnetic charges and arbitrary electric charges [3]. He proved

this by using the moduli space of the 2 monopoles and its hyperkahler structure.

Define the r monopole moduli space Mr as the following manner,

Mr = R3 × S1 ×M0r

Zr, (1.91)

29

R3 means the motion of The center of mass of r monopoles. S1 represents the χ

coordinate conjugate to the total charge of the monopole, and M0r is the internal

moduli space with 4(r − 1) coordinates. Zr is generated by g as the shifting χ →

χ + 2πr and a non-trivial action on M0

r . For the quantum bound states of dyons

can be identified as Z-invariant harmonic 2-form on S1×M0r . For the dyons with p

electric charge, the wavefunction should be proportional to eipχ, g transformation

generates the additional phase e2πipr from the eipχ. Therefore, the wavefunction on

M0r should generate e−

2πipr phase from g transformation.

In [14], it was pointed out that in the supersymmetric system, the BPS states

has one-to-one mapping to the harmonic form in the moduli space. Since SL(2,Z)

duality require that the harmonic form for the BPS dyons should be unique on M0r ,

which has d = 4(r−1) dimensions. The Poincare duality implies that normalizable

n-forms and the hodge dual d−n-form exist pairwise. Therefore, the harmonic form

for the BPS dyons should be the unique (anti-)self 2(r−1)-form. In [3], Ashoke Sen

pointed out that SL(2, Z) invariance requires the existence of the (anti-)self-dual

harmonic 2(r − 1)-form on M0r acquiring a phase e−

2πipr under g transformation

of the generator Zr. Moreover, he explicitly constructed the normalizable self-dual

harmonic 2-form with r = 2 on M0r .

The 2-monopole internal moduli space, M0r=2 is the Atiyah-Hitchin space [15]

whose metric is given by,

ds2 = f2dρ2 + a2(σ1)2 + b2(σ2)

2 + c2(σ3)2 . (1.92)

where 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π, and 0 ≤ θ ≤ 2π. with the following identification,

(ρ, θ, ϕ, ψ) ≡ (ρ, π − θ, π + ϕ,−ψ). (1.93)

30

f , a, b, and c are some known function of ρ. σ1,2,3 are defined by,

σ1 = − sinψdθ + cosψ sin θdϕ ,

σ2 = cosψdθ + sinψ sin θdϕ

σ3 = dψ + cos dϕ . (1.94)

These hold the following identity,

dσi =1

2ϵijkσj ∧ σk . (1.95)

The stable dyons with 2 monopole charge should have the odd number of the

electric charges, and the self-dual 2-form for these dyons should acquire the extra

minus sign, e−2πiµe

2 = −1, under Z2 transformation, which is given by

(ρ, θ, ϕ, ψ) → (ρ, θ, ϕ, ψ + π). (1.96)

In this space, Ashoke Sen found the unique (anti-)self-dual 2-form ω for these dyons

with (µe, 2) charge given by

ω = F (ρ)(dσ1 −

fa

bcdρ ∧ σ1

)(1.97)

where F (ρ) is given by

F (ρ) = F0 exp

(−∫ ρ

π

fa

bcdρ′). (1.98)

where F0 = F (ρ). Moreover, he proved this self-dual 2-form is smooth and normal-

izable on M0r=2. This results means that there exists the unique quantum bound

states of dyons with 2 monopole charges and odd number electric charges.

31

1.2.3 Seiberg duality in 4d N = 1 Super QCDs

While 4d N = 4 supersymmetric gauge theories have SL(2,Z) electromagnetic

duality, it has been suggested that two different 4d N = 4 supersymmmetric QCDs

can describe the same IR fixed point, and related to each other by electromagnetic

duality. This duality is called ‘Seiberg duality’ [16].

Note that the exact beta function of N = 4 supersymmetric QCD with SU(Nc)

gauge group and Nf flavors is given by,

β(g) =g3

16π23Nc −Nf +Nfγ(g

2)

1−Ncg2

8π2

(1.99)

γ(g2) is the anomalous dimension of the mass, which is given by

γ(g2) = − g2

8π2N2c − 1

Nc+O(g4) . (1.100)

For Nf ≥ 3Nc, it can be shown that these theories are not asymtotically free.

Moreover, the gauge coupling of this theory become weaker at long distance by

screening quantum effects. these theories can be a consistent interacting quantum

theory due to the Landau pole in UV. They can be useful only as the low energy

effective theories.

On the other hand, for 32Nc < Nf < 3Nc cases, there exist IR fixed points

for these supersymmetric QCDs, and these IR fixed points describe the interacting

superconformal field theories. These fixed points are kinds of the Bank-Zaks fixed

points [17]. In a infinite limit of Nf and Nc with fixingNf

Nc= 3−ϵ, one can construct

the IR fixed points explicitly,

Ncg2∗ =

8π2

3ϵ+O(ϵ2) (1.101)

For finite Nf and Nc, existence of the IR fixed points is guaranteed by superconfor-

mal structure and its unitary bound. In these 4d superconformal field theories, the

external electric and magnetic sources interact with each other by the potential,

32

which is given by V ∼ 1R . Therefore, these IR superconformal field theories are in

non-abelian Coulomb phase.

Spectrum of the superconformal field theories is given by the gauge invariant

local operators, by operator-state correspondence in conformal field theory. N = 1

Superconformal algebra give the severe constraints on the spectrum of quantum

states. For example, 4d N = 1 superconformal field theories should hold the fol-

lowing bound for scalar fields.

D ≥ 3

2|R| . (1.102)

D is the conformal dimension of the local operators, and R is R-symmetry charge

involved in superconformal algebra. The local operators which saturate the above

bound are called chiral primary operators. In superconformal case, the scalar com-

ponents of the gauge invariant chiral with D = 32R and anti-chiral superfields with

D = −32R saturate the unitary bounds.

4d N = 1 supersymmetric QCDs have SU(Nf )×SU(Nf )×U(1)R symmetry,

and the massless degrees of freedom of these theories are given by mesonic and

baryonic operators, defined by

M = QQ , (1.103)

B = Qi1Qi2 ·QiNc ϵi1i2···iNc, (1.104)

Since the elementary chiral superfield in the UV gauge theory hasNf−Nc

Nf, The

conformal dimension of the meson fields and the baryonic fields can be estimated

by,

D(M) = D(QQ) ≡ 2 + γ∗ =3

2· 2 ·

Nf −Nc

Nf=

2(Nf −Nc)

Nf. (1.105)

D(B) =3Nc(Nf −Nc)

Nf. (1.106)

With this result, the anomalous dimension is given by

γ∗ = 1− 3Nc

Nf(1.107)

33

SU(Nf −Nc) SU(Nf ) SU(Nf ) U(1) U(1)R

q N Nf 1 NcNf−Nc

NcNf

q N 1 Nf − NcNf−Nc

NcNf

Dual meson 1 Nf Nf 0 2Nf−Nc

Nf

Table 2: Field contents and charges in dual theories for 4d N = 1 QCDs

In this point, the gauge coupling (1.99) vanishes, and the IR fixed point is guaran-

teed.

The unitary bound coming from conformal algebra constrains spectrum of the

scalar fields, as follows,

D(ϕ) ≥ 1 . (1.108)

From (1.105), the following inequality can be obtained,

Nf ≥ 3

2Nc . (1.109)

Therefore, in 32Nc ≤ Nf ≤ 3Nc, there exist the IR fixed points of 4d N = 1 super-

symmetric QCDs. The range of 32Nc ≤ Nf ≤ 3Nc is called ‘Conformal window’.

4d N = 1 supersymmetric QCDs have the dual theories [16]. These original

theories and the dual theories are different as UV theories, however, they should

describe the same IR fixed points, and this fact has been supported by non-trivial

results. This duality is called ‘Seiberg duality’. This duality has aspects of electro-

magnetic duality. Seiberg duality has been regarded as the example of electromag-

netic duality which holds quantum mechanically as a more realistic case.

The dual theories have Nf flavors and the SU(Nf − Nc) gauge group, and

involve a dual meson field. The charges of the dual quarks and mesons are given

by Table. 2.

These dual theories involve the following superpotential,

W =1

µM i

jqiqj . (1.110)

34

Since the dual theories are in the conformal window, 32(Nf −Nc) < Nf < 3(Nf −

Nc), these theories have the IR conformal fixed points.

Seiberg duality has the non-trivial checks. Firstly, the original theories and

the dual theories have the same anomaly structure, and hold the ‘t Hooft anomaly

matching condition. Secondly, these theories have the same moduli space. The

mesonic fields as composite fields of the quarks in the original theories is matched

with the dual mesons in the dual theories. The baryonic spectrum can be directly

matched each other. Moreover, with the flavor mass terms and integrating out the

massive flavors, the duality structure is preserved.

The beta function of these dual theories is given by

β(g) ∝ −g3(3(Nf −Nc)−Nf ) = −g3(2Nf − 3Nc) (1.111)

Therefore, these Seiberg dual theories lose their asymtotic freedom, when Nf <

32Nc. one can call this dual theories as ‘magnetic dual theories’, and the original

theories as ‘electric theories’. In the conformal window, as the electric coupling

become stronger, the coupling in magnetic dual theories become weaker. This is

analogous to g → 1g in electromagnetic duality. Outside of the conformal window,

it has been conjectured that the electromagnetic duality holds and exchanges the

free electric phase and free magnetic phase.

Recently, 3d Seiberg-like dualities has been proposed and supported by various

non-perturbative studies [18, 19]. 3d Seiberg duality is similar to 4d Seiberg duality

in that these dualities exchanges the gauge group from SU(Nc) to SU(Nf −Nc).

However, physical interpretation is very different. The gauge coupling in 3d gauge

theories is relevant, and these theories are always strongly-coupled in IR. Therefore,

3d Seiberg-like duality are always strong-strong coupling duality. One of the main

examples of 3d Seiberg duality in 3d N = 4 SUSY gauge theories will be studied

in this thesis, via the non-perturbative dynamics of the solitonic vortices.

35

Chapter 2

Little strings and T-duality

2.1 Introduction

The result of this chapter is based on the paper [20].

Little string theories are non-critical string theories on lower dimensional

spacetime than critical superstring theories on 10 dimensional spacetime. Exis-

tence of various 6d little string theories has been predicted from decoupling limits

of fundamental strings near N NS5-branes, which exist in critical superstring the-

ories. For example, little string theories with 16 supercharges can be obtained from

type II fundamental strings bound to N NS5-branes. These are called ‘type II

little string theories’. Type IIA little string theories from IIA NS5-branes have 6

dimensional N = (2, 0) super-poincare symmetry, while type IIB little string theo-

ries from IIB NS5-branes have N = (1, 1) super-poincare symmetry. The heterotic

strings bound to N NS5-branes also indicate that there exist two types of ‘Heterotic

little string theories’. Depending on whether the heterotic strings involve SO(32)

or E8×E8 gauge symmetries, there are two type of ‘heterotic little string theories’

with SO(32) or E8×E8 global symmetries. They have 6d N = (1, 0) super-Poincare

symmetry. In this thesis, I will focus to study type II little string theories.

Especially, type II little string theories are interesting because of the following

reasons. type IIA little string theories have the low energy limit described by 6d

N = (2, 0) superconformal field theories. For IIB cases, the low energy dynamics of

IIB NS5-branes is governed by 6dN = (1, 1) maximally supersymmetric Yang-Mills

theories, of which gauge interaction is non-renormalizable and involve UV-cutoff

scale. On much higher energy scale than the cutoff scale, type IIB little string

theories can be regarded as UV-completion of the 6d maximal super-Yang-Mills

36

theories. Since NS5-branes are one of the most difficult non-perturbative objects

to study, the detail of the little strings’ dynamics is still mysterious.

Type II little string theories have similarities and differences comparing with

critical string theres. The fact that these little string theories should not involve

gravitational interaction is a main difference. On the other hand, as non-local the-

ories, the little strings inherit many stringy properties from type II fundamental

strings. For example, type II little string theories have interesting feature of T-

duality. With circle-compactifying the little string theories, they are supposed to

T-dual each other. One of the main purpose of this chapter is to clarify aspects

of T-duality between type II little string theories, for similarities and differences

comparing with 10d critical type II superstring theories.

In comparing with critical strings, the noncritical little strings can not be

easily studed for several reasons. For critical strings, their strength of interaction is

usually tuned by the string coupling constant gs. If we have a small gs, we can study

dynamics of critical strings with perturbative ways. However, little string theories

have only one physical parameter of the string length scale ℓs, and have no coupling

constant. Therefore, they are intrinsically strongly-coupled. Some studies of them

use : holography near NS5-branes [21, 22], discrete lightcone quantization(DLCQ)

[23, 24, 25, 26], the double scaling limit [27, 28].

Especially, the DLCQ of the little string theories is realized by compactification

of the little string theories on a small circle. In this limit, the quantized momentum

on the circle will be highly massive and decoupled from the rest. As the distinct

sectors of the definite number of the momenta, we can describe dynamics of them.

This is equivalent to a large circle limit of the T-dual little string theory, in which

we keep the definite winding number of the T-dual little strings. Since the winding

T-dual little strings involves infinite towers of momentum on the large circle, the

DLCQ of the little string theories are given as 2 dimensional quantum field theories,

describing world-sheets of T-dual winding little strings.

In this chapter, I will study 2d QFT descriptions of macroscopic little strings.

Macroscopic little strings are infinitely extended strings on R1,1 and they are highly

37

massive. 2d degrees of freedom which decoupled from the 6d bulk and localized on

the strings are described by these 2d QFTs. These theories are studied in [25, 26].

I study the T-duality of little strings in the BPS sector, from the UV gauge theory

descriptions. In particular, being able to compute the elliptic genus indices on both

IIA and IIB sides, thanks to our new gauge theory descriptions, their BPS spectra

can be directly compared. I find, in fugacity expansions to highly nontrivial or-

ders, that the two elliptic genera precisely map to each other via T-duality.1 Apart

from confirming the naturally expected T-duality, our finding is establishing a very

nontrivial identity between the elliptic genera computed from the type IIA and

IIB sides, so that alternative expressions can be used to extract various properties

which would have been very difficult to see from the other viewpoints. For instance,

I explain in section 5 how one can easily understand the SL(2,Z)×SL(2,Z) trans-

formation properties of the elliptic genus, for the complex structure and Kahler

parameters of the torus, by using our T-dual expressions for the elliptic genus.

The rest of this chapter is organized as follows. In sections 2 and 3, I explain

the 2d gauge theory descriptions of the IIB and IIA little strings, respectively, and

study their elliptic genera. In section 4, we study the T-duality of the two elliptic

genera, as well as extended duality/triality properties. In section 5, we study the

SL(2,Z) transformation properties of the elliptic genus in various fugacities. Section

6 concludes with brief discussions.

2.1.1 DLCQ description of type II little string theories

Lightcone quantization has been a very useful method to study quantum field

theories, especially in studying scattering of hadrons in the collider physics. The

lightcone quantization starts from defining the following lightcone coodinate,

X± =x0 ± x1√

2, X2,··· ,d−1 = x2,··· ,d−1 . (2.1)

1In order to better define our spectral problem, without continua coming from the ‘throat’regions [25, 29, 26], I turn on the Fayet-Iliopoulos (FI) term and the theta angle of the gaugetheories on the worldsheet. Also, to avoid having infrared problems with tensionless fractionalstrings or W-bosons, I separate the N NS5-branes and study the massive spectra.

38

The contractions between two vectors is given by

v · w = −v+w− − v−w+ +d−1∑i=2

viwi . (2.2)

In this coodinate, X+ is interpreted as a new time coordinate, and X− as a new

spatial coordinate. With this interpretation, p+ has a role of Hamiltonian, and p−

is a conserved quantity. For a particle dynamics, the following identity holds,

− 2p+p− + |p⊥|2 = −M2. (2.3)

Therefore, we can obtain

p+ =|p⊥|2

2p−+M2

2p−. (2.4)

This means that the lightcone description of the system give a non-relativistic en-

ergy and a potential term, proportional to 1p− . Applying this framework to quantum

field theory enable to resolve the complexity to describe the detail of dynamics and

many subtleties in quantum field theories.

There are many ways to obtain non-relativistic limits of the physical system.

For example, if we take a spatial direction of x1, and infinitely boost the negative

direction of x1. This frame is called the infinite momentum. In this frame, the

physical modes with a finite p1 should be integrated out, and we can expect a

non-relativistic effect of the system. However, it is usually difficult to obtain a non-

relativistic limit in the infinite momentum frame, due to many technical problems

and subtleties.

However, the lightcone description of physical system provide one of the eas-

iest way to obtain a non-relativistic description of the system with the Galilean

invariance. And, this description is well-defined with every finite p−.

A variation of the lightcone quantization is circle-compactifying the direction

39

of x−. In this situation, the lightcone momentum, p−, is quantized by,

p− =N

R−, N ∈ Z (2.5)

where R− is the radius of the lightcone direction. Since the p− is a conserved

quantity, the physical system with a definite momentum number N is always well-

defined. We can consistently obtain the non-relativistic quantum mechanics of the

quantized N lightcone momenta, regardless of the radius, R. This is called discrete

lightcone quantization(DLCQ).

Alternatively, if compactifying the spatial direction of x1, the p1 momentum

is quantized as N/R1. the circle-compactified infinite momentum frame is not well-

defined in several ways. Firstly, since p1 is not a conserved quantity, we can not

expect non-relativistic effects with a finite p1. Only in p1 → ∞ limit, we can expect

non-relativistic effects and describe a non-relativistic dynamics of decoupled infinite

p1 quantized momenta. Boosting to obtain a large p1 momentum is equivalent to

shrink the radius of the x1. This means that this non-relativistic limit only holds in

a small R1, while the discrete lightcone quantized system provides a non-relativistic

quantum mechanics description of system with all R−. Usually, obtaining this non-

relativistic quantum mechanics description in the infinite momentum frame is much

more difficult than in the discrete lightcone quantization due to complexity of

detail of dynamics and subtleties. Moreover, there is not any reason that this non-

relativistic description in the infinite momentum frame should be coincident with

one in the discrete lightcone quantization.

However, if the physical system has space-time supersymmetry, supersym-

metry imposes severe constraints the detail of interaction of the system. There-

fore, we can expect the discrete lightcone quantized system may provide the same

non-relativistic description of the dynamics of the decoupled quantized momenta

p1 = N/R1 in the small R− limit. This has been conjectured, and supported by

various non-trivial evidence [23, 30, ?].

Usually, the DLCQ description of quantum field theories provides a non-

40

relativistic quantum mechanics. However, DLCQ description of string theories pro-

vides 2d field theories on a circle, since the discrete lightcone quantized theory

keeps finite lightcone momenta with involving infinite towers of the winding strings

on a circle, which provides 2d field theory degrees of freedom.

As I already mentioned, DLCQ descriptions of supersymmetric systems can be

interpreted as effective quantum mechanics description of the quantized momen-

tum on a small spatial circle. In T-dual picture, 2d supersymmetric field theories

as DLCQ descriptions of superstring theories can be regarded as the effective de-

scription of the macroscopic strings winding the a large spatial T-dual circle. I will

use the DLCQ descriptions as 2d field theories to study dynamics of macroscopic

strings in type II little string theories.

DLCQ description of IIA little string theory

IIA little string theory is derived from the gravity decoupling limit of the

fundamental strings on type IIA NS5-branes. 2d field theory description of DLCQ

type IIA little string can be constructed, as follows [26].

Firstly, the DLCQ type IIA string theory with p− = kR−

momentum is de-

scribed by 2d U(k) N = (8, 8) maximally supersymmetric gauge theory. This can

be equivalent to the gauge theory description of T-dual IIB macroscopic strings,

which is under S-dual relation to D-strings type IIB string theory. The gauge cou-

pling is given by

gYM =R−gsℓ2s

(2.6)

In considering IIA little string theory, N of type IIA NS5-branes provides an ad-

ditional N = (4, 4) hypermultiplet charged by SU(N) global symmetry and U(k)

gauge symmetry. This DLCQ description of IIA little string theory has a direct

relevance with the macroscopic IIB little strings. This gauge theory action is given

in Appendix A.

Actually, the DLCQ type IIA little string theory is the conformal field theory

obtained in IR of the 2d N = (4, 4) gauge theory description. This gauge theory has

41

U(k)N

U(k)2U(k)1

U(k)3

U(k)4

Figure 1: AN−1 quiver diagram of the 2d N = (4, 4) gauge theory for the DLCQIIB little string theory. Dashed lines denote the hypermultiplets.

two decoupled superconformal field theories, whose target spaces are the Coulomb

branch moduli space or Higgs branch moduli space. Since the Higgs branch moduli

space describes the motion on the NS5-branes, as the instanton moduli space, the

Higgs branch conformal field theories describes the DLCQ type IIA little string

theory. On the other hand, the Coulomb branch describes the motion coming out

of the NS5-branes, and it is irrelevant to the little string theory.

DLCQ description of IIB little string theory

The DLCQ description of IIB little string theory is derived from gs → 0 limit of

the theory on AN−1 singularity in type IIA string theory. If T-dualizing this system,

we can obtained the k fundamental strings near the similar singularity in type IIB

string theory. It can be naturally S-dualized to D-strings near the singularity. From

this viewpoint, we can obtain the 2d N = (4, 4) gauge theory description of the

DLCQ IIB little string theory.

This gauge theory has U(k)N gauge group and forms the circular quiver dia-

gram of Figure. 1. On each node, U(k)i N = (4, 4) vector multiplets are assigned,

and N = (4, 4) bi-fundamental hypermultiplets charged under U(k)i×U(k)i+1 are

introduced. This 2d gauge theory has also the decoupled Coulomb branch SCFT

42

and Higgs branch SCFT. Contrary to the DLCQ type IIA little string theory, The

Coulomb branch SCFT is relevant to the DLCQ IIB little string theory. This gauge

theory has SO(4)× SO(3)R global symmetry in UV, where SO(3) is the isometry

of AN−1 singularity. In IR, we can expect that SO(3)R symmetry is enhanced to

SO(4)R symmetry, and SO(4)R symmetry is a true symmetry of IIB little string

theory. Details of this theory will be studied more in section 3.

In studying the IR spectrum in the DLCQ IIB string theory by using this

gauge theory, there are some restrictions. Since this gauge theory has only SO(3)R

instead of SO(4)R, we can not track charge information for the Cartan elements of

the global symmetry. This will be cured by introducing a new 2d gauge description

in section 3.

Throat regions as Continuum from UV, and Holography description

The 2d N = (4, 4) gauge theory description of the DLCQ type IIA little string

theory classically has the decoupled Coulomb branch and Higgs branch. However,

near the origin of the Higgs branch, the quantum mechanical branch as a infinite

throat region is developed to connect near the origin of the Coulomb branch. This

throat region is interpreted as a UV continuum spectrum. In [26], the origin of this

throat region is explained in detail. With a semi-classical approach for k = 1 case,

the metric of the throat region and torsion can be obtained as,

ds2 = dϕ2 +N

2dΩ2

3 , H = −NdΩ3 . (2.7)

The central charges of the conformal field theory with the target space of the throat

region can be obtained as,

Q = (N − 1)

√2

N. (2.8)

The low energy spectrum in the throat region can be given by,

HDLCQ =R−ℓ2s

(p2 + q(q + iQ) + 1− N

2

)(2.9)

43

where q is the conjugate momentum of ϕ.

This throat region has a definite physical meaning in the holographic descrip-

tion [21, 26]. The near horizon geometry of type II NS5-branes is the linear dilaton

backgrouds, and includes this throat region. The near horizon geometry of type II

NS5-branes is given by,

ds2 = ds2(R1,5) + dϕ2 +N

2dΩ2

3 , H = −NdΩ3 . (2.10)

The string coupling behaves like

gs ∝ e−ϕ

√2N . (2.11)

The string worldsheet theory on this space consists of a free scalar field with a

background charge of Q =√

2/N , four free fermions, a bosonic level (N − 2)

SU(2) Wess-Zumino-Witten model, (and six free scalars and fermion describing

the motion on R1,5).

In this situation, the low energy spectrum in (2.9) on the throat region in

DLCQ model corresponds to the continuum spectrum coming from the supergravi-

tons and the long strings with the string coupling of gs = e−ϕ

√2N in the holographic

approach of the near horizon geometry of the NS5-branes [21, 26].

2.2 IIB little strings

2.2.1 A brief review

I first consider the type IIB little strings, which are the type IIB fundamental

strings bound to the NS5-branes. At low energy, the world-volume description of

IIB NS5-branes is given by 6d maximally supersymmetric Yang-Mills theory, with

(1, 1) supersymmetry and U(N) gauge group. The fields consist of the gauge field

Aµ=0,··· ,5, 4 scalar fields ϕI=1,··· ,4, and fermions. These degrees are provided by

the D-strings ending on the NS5-branes. The bosonic symmetry of the theory is

44

SO(1, 5) × SO(4)R. SO(1, 5) is the Lorentz symmetry on the NS5-branes, and

SO(4)R is the symmetry on their transverse directions, which rotates ϕI . The

Yang-Mills coupling constant is given by

g2YM =1

TNS5(2πα′)2g2s= (2π)3α′ . (2.12)

Fundamental strings form threshold bounds with the NS5-branes. They are identi-

fied as the instanton strings in the 6d SYM. The instanton string tension is given

by4π2

g2YM

=1

2πα′ = TF1 , (2.13)

agreeing with the tension of the fundamental string. The coupling constant is in-

dependent of the 10d string coupling constant, gs. So one can take the little string

theory limit, gs → 0 with fixed α′. All the gravitational degrees of freedom are

decoupled.

I shall consider k macroscopically extended little strings, extended along R1,1

part of R5,1. I am interested in the dynamics of the degrees of freedom supported on

these macroscopic strings, decoupled from the rest of the 6d degrees at low energy.

The system of k F1 and N NS5-branes admit a UV gauge theory description given

by a U(k) gauge theory withN = (4, 4) supersymmetry. The field theory is identical

to that living on the D1-D5 system via S-duality, and has been studied extensively

in the literature, e.g. [25, 26, 29]. This 2d theory at low energy can also be regarded

as the worldsheet description of the instanton strings of the 6d SYM theory. The

gauge theory has the U(k) N = (4, 4) vector multiplet, an adjoint hypermultiplet,

and N fundamental hypermultiplets which host U(N) global symmetry. These

fields are shown in Table 3, and more details about this theory is explained in

Appendix A. For later convenience, we also show the supermultiplet structure with

respect to the right-chiral (0, 4) SUSY. The bosonic symmetry preserved by the

strings is SO(1, 1)×SO(4) ⊂ SO(1, 5) times SO(4)R, where the latter is inherited

from the R-symmetry of the 6d theory. For SO(4) ∼ SU(2)L1 × SU(2)R1 and

45

N = (4, 4) N = (0, 4) Fields U(k) U(N)

vector vector Aµ, λAα+ adj 1

twisted hyper φaA, λαa− adj 1

hyper hyper aαβ, λAα− adj 1

Fermi λaβ+ adj 1

hyper hyper qα, ψA− k N

Fermi ψa− k N

Table 3: N = (4, 4) supermultiplets for k IIB strings.

SO(4)R ∼ SU(2)L2 × SU(2)R2, I introduce the following doublet indices,

SU(2)L1 → α, SU(2)R1 → α, SU(2)L2 → a, SU(2)R2 → A . (2.14)

The fields in Table 3 and appendix A are given with this convention. The 6d (1, 1)

supercharges can be written as Qaα+, QAα−, Q

αa+, Q

Aα− , where ± denote 6d chirality.

These supercharges satisfy the reality conditions given by

Qaα+ = −ϵαβϵab(Qbβ+)†, QAα− = ϵαβϵAB(QBβ−)

†

Qαa+ = −ϵαβϵab(Qβb+)†, QAα− = ϵαβϵAB(QBβ− )†. (2.15)

The strings extended on R1,1 preserve Qαa+ and QAα− , forming 2d N = (4, 4) su-

persymmetry. The ± subscripts on 2d fermions denote left/right chiralities, re-

spectively. In Table 3, the (4, 4) Higgs branch fields aαβ and qα form the so-called

ADHM data of k multi-instantons of U(N) gauge theory. This is because the IR

dynamics of this gauge theory will be describing the 6d instanton strings, as I shall

explain in more detail now.

The infrared dynamics of this (4, 4) theory has been studied in [25]. Its low

energy dynamics is described by two decoupled (4, 4) conformal field theories. One

is the conformal field theory on the Higgs branch described by a nonlinear sigma

model on the Higgs branch target space, given by k instanton moduli space. Another

is the conformal field theory on the Coulomb branch. For studying the type IIB

46

little strings, the Higgs branch CFT is of relevance. The Coulomb branch degrees

φaA represent the motion of the strings moving away from the 5-branes.

There is a peculiar singularity in the region near qα = 0, aαβ = 0, where

the Higgs branch classically meets the Coulomb branch [25, 26, 29]. Quantum

mechanically, this region forms a ‘throat,’ which is responsible for a continuum in

the CFT spectrum. The CFT can be deformed by turning on the SU(2)R1 triplet

of Fayet-Iliopoulos term ζI (I = 1, 2, 3) and the theta angle θ, after which the

last continuum disappears. In particular, the Higgs branch moduli space becomes

regular, and the Coulomb branch is no longer connected to the Higgs branch even

classically. I shall consider the little string spectrum with nonzero FI term, from

the elliptic genus index [31] of the gauge theory. (The continuum will be completely

lifted, not only by the FI-term but also with the Coulomb VEV of the 6d SYM

to remove the infrared continuum.) In particular, with nonzero FI parameter, the

elliptic genus will acquire contribution only from the Higgs branch CFT for the IIB

little strings, and not from the Coulomb branch CFT that I am not interested in.

2.2.2 The elliptic genus of IIB little strings

In this subsection I shall define and explain the elliptic genus of the gauge theory

compactified on circle, counting 14 -BPS states in the Coulomb phase of the 6d

theory, which shall be further studied in sections 4 and 5. This is a supersymmetric

partition function on a torus with complex structure τ . I choose a supercharge

Q = QA=1,α=2 and define its index, with q ≡ e2πiτ ,

ZIIBinst(αi, ϵ±,m; q, w)

= Tr[(−1)FwkqHL qHRe2πiαiΠie2πiϵ−(2J1L)e2πim(2J2L)e2πiϵ+(2J1R+2J2R)

](2.16)

P is the momentum on the string compactified on the circle, and H is the energy,

in the unit of inverse-radius R−1B of the circle. 2HL = H + P and 2HR = H − P

are defined as the leftmoving and rightmoving momentum, respectively. JL1,2 and

JR1,2 are the Cartans of SU(2)L1,2 and SU(2)R1,2. Since Q,Q† = 2HR and Q

47

commutes with all the other factors in the trace, the index counts only the BPS

states annihilated by Q and Q†, and it is independent of q. Πi’s are the Cartans

of U(N). αi’s are the chemical potentials for electric charges, interpreted as the

background gauge field A5 = diag(αi) along the spatial circle, breaking U(N) to

U(1)N . I also introduce the fugacity variable, w, counting the winding number k

of the little strings. For a given U(k) gauge theory, I fix k. The above index is the

grand partition function. I use the subscript ‘inst’ (standing for instantons) in the

6d SYM interpretation, as I have already explained that the gauge theory index

will acquire contributions only from the Higgs branch.

It is also useful to consider the full index of the type IIB little string theory,

compactified on a circle with large radius RB ≫ (α′)12 . The index is defined in the

same way as (2.16), where the trace is taken over the whole BPS Hilbert space of

the 6d theory in the Coulomb phase. This is a BPS partition function on R4 × T 2.

Apart from (2.16), one finds extra contribution from the 6d perturbative SYM

states, which are decoupled with the winding strings at low energy. The full index

thus factorizes as

ZIIB(αi, ϵ±,m; q, w) = ZIIBpert(αi, ϵ±,m; q)ZIIB

inst(αi, ϵ±,m; q, w). (2.17)

The 6d perturbative index, ZIIBpert, counts the modes which only carry momenta

along the circle. This can be computed as follows. The momentum along the circle

preserves supercharges Qaα+ and QAα− , and breaks Qαa+ and QAα−. The Goldstino

zero modes coming from the broken SUSY generators contribute to the single par-

ticle index with the following factor,

24 sinh2πi(m+ ϵ+)

2sinh

2πi(m− ϵ+)

2sinh

2πiϵ12

sinh2πiϵ22

, (2.18)

and the bosonic zero modes on R4 provides the factor

1

24 sinh2 2πiϵ12 sinh2 2πiϵ2

2

(2.19)

48

where ϵ± ≡ ϵ1±ϵ22 . Therefore, the single particle index of the particle carrying KK

momentum is given by [32],

I+(ϵ±,m) =sinh 2πi(m+ϵ+)

2 sinh 2πi(m−ϵ+)2

sinh 2πiϵ12 sinh 2πiϵ2

2

. (2.20)

The single particle index of the 6d perturbative particles is given by

zsp =NI+(ϵ±,m) ·∞∑n=1

qn

+ I+(ϵ±,m) ·

N∑i>j

e2πi(αi−αj) +N∑i =j

∞∑n=1

e2πi(αi−αj)qn

= I+

N∑i>j

e2πi(αi−αj) + I+

N∑i,j

e2πi(αi−αj)

q

1− q. (2.21)

From this, ZIIBpert is given by

ZIIBpert(αi, ϵ±,m; q) = PE

[zsp(αi, ϵ±,m; q)

]= exp

∞∑p=1

1

pzsp(pαi, pϵ±, pm; qp)

.

(2.22)

The contribution of the winding IIB little strings ZIIBinst is given in terms of the

elliptic genera Zk of the k instanton strings by

ZIIBinst(αi, ϵ±,m; q, w) =

∞∑k=0

wkZk(αi, ϵ±,m; q) , (2.23)

where Zk=0 ≡ 1. Zk is given by the sum of the terms characterized by N -colored

Young diagrams, Y = Y1, Y2, · · · , YN . The sum of the numbers of the boxes∑Ni=1 |Yi| is k. The elliptic genus is given by [33, 34]

Zk(αi, ϵ±,m; q) =∑

Y :∑

i |Yi|=k

N∏i,j=1

∏s∈Yi

θ1 (q;Eij +m− ϵ−) θ1 (q;Eij −m− ϵ−)

θ1 (q;Eij − ϵ1) θ1 (q;Eij + ϵ2),

(2.24)

49

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x11(S1M)

N M5 × × × × × × αini M2 × × (αi, αi+1)

Table 4: M-theory brane uplift of IIA little strings.

where

Eij = αi − αj − ϵ1hi(s) + ϵ2vj(s). (2.25)

‘s’ denotes a box in the Young diagram Yi. hi(s) is the distance from the box ‘s’

to the edge on the right side of Yi that one reaches by moving horizontally. vj(s)

is the distance from ‘s’ to the edge on the bottom side of Yj that one reaches by

moving vertically. See e.g. [32] for more details and illustrations. The expression

(2.24) may be computed by the contour integration formula given in terms of the

Jeffrey-Kirwan residues [35, 36, 37], as explained in [38].

2.3 IIA little strings

Type IIA NS5-branes realize 6d IIA little string theory, with N = (2, 0) supersym-

metry. The light degrees should be made of Bµν world-volume tensor gauge field

whose field strength is self-dual in 6d, and 5 scalars, ϕI=1,2,3,4 and ϕ, and fermions.

ϕI=1,2,3,4 parametrize the transverse R4 of type IIA string theory, and ϕ is a com-

pact scalar parametrizing the position of the 5-branes along the M-theory circle.

The little strings are type IIA fundamental strings bound to the NS5-branes. In M-

theory, type IIA fundamental strings uplift to M2-branes wrapping the M-theory

circle. The limit gs → 0 with a fixed α′ yields the N = (2, 0) little string theory.

The 2d little string gauge theory valid at RA ≫ (α′)12 has been studied in

[26], in the ‘Coulomb phase’ with nonzero ϕ, separating all M5-branes along x11.

[26] discussed it in the context of type IIB strings on AN−1 singularity, but let us

review it in the M-theory context here. The M-theory branes are shown in Fig. 2,

where the M-theory circle radius is given by RM = gsℓs (where α′ = ℓ2s). See also

50

x01234

x11

x5

M5 M5 M5

M2 M2 M2 M2

Figure 2: M-theory brane uplift of the IIA little strings

Table 4 for coordinates. The tension of the strings is given by ∼ RM

ℓ3P= RM

gsℓ3s= ℓ−2

s

in the original type IIA string theory, and I am interested in the low energy 2d

theory at excitation energy E ≪ gYM, where gYM is the 2d gauge coupling scale.

To ease the construction of this theory, we compactify x9 direction along a circle

with radius R′M . Since NS5-branes are localized at x9 = 0 and M2-branes are

attached to them, this compactification cannot be seen by the low energy CFT on

the strings, although it will be seen by the UV gauge theory I construct. Now I

make a 9-11 flip, regarding x9 as the M-theory circle direction. The new type IIA

theory would have its own coupling and string scale g′s, ℓ′s, satisfying g

′sℓ

′s = R′

M ,

g′s(ℓ′s)

3 = ℓ3P . The tension of the string given by the D2-branes suspended between

NS5-branes is RMg′s(ℓ

′s)

3 = RM

ℓ3P= ℓ−2

s , same as in the original type IIA picture. Now

the low energy 2d theory living on the D2-branes is easy to identify. It is a circular

quiver U(k)N gauge theory with N = (4, 4) supersymmetry [39]. Each gauge node

(labeled by i = 1, · · · , N) has vector multiplet fields A(i)µ , a

(i)

αβand fermions, where

α, β are the SO(4) = SU(2)L1×SU(2)R1 spinor indices. There is a bi-fundamental

hypermultiplet mode connecting adjacent gauge node, and between i’th and i+1’th

51

node, the fields are denoted by complex scalars Φ(i)A and fermions. Compared to be

previous type IIB setting, or the original type IIA setting, in which I had SO(4) =

SU(2)L2 × SU(2)R2 R-symmetry, only the diagonal SU(2)D survives after the x9

circle compactification. So the doublet A index can be regarded as the identification

of the previous a and A indices. The (4, 4) supercharges are QAα+, QAα−, subject to

reality conditions. The SU(2)D UV symmetry is supposed enhance to full SO(4) in

IR, but is invisible in UV. The incapability of seeing the second Cartan of SO(4)

from this UV theory will make it impossible to study the full IR elliptic genus.

This will be a motivation to study a (0, 4) supersymmetric UV gauge theory for

the type IIA little strings, in section 3.1.

The coupling for the i’th U(k) gauge field is given by

1

g2YM,i

=(αi+1 − αi)RMℓ

′s

g′s=

(αi+1 − αi)g2sℓ

4s

(R′M )2

, (2.26)

which become infinite in the little string decoupling limit gs → 0. All these couplings

become large in the IR limit on the strings E ≪ gYM,i. One can turn on three FI

parameters ζ(i)I for each U(k)i gauge group, which is a triplet of SU(2)D rotationg

678. This corresponds to the relative position of the i + 1’th NS5-brane from the

i’th NS5-brane along 678 directions. So one obtains the condition∑N

i=1 ζ(i)I = 0,

since one should come back to the original NS5-brane after going around the quiver

once.

The gauge theory has U(k)N Coulomb branch, whose scalars represent the

motion of D2-branes along 1234 directions. This would define the Coulomb branch

CFT which is relevant for studying the IIA little strings. On the other hand, the

N fractional strings suspended between different adjacent pairs of NS5-branes can

combine to make a fully winding D2-brane along x11, which may leave the NS5-

brane along the 6789 directions (among which x9 is the circle direction of the

M-theory). For instance, at k = 1, the positions of the D2-branes along 678 is

parameterized by the Higgs branch scalars, breaking U(1)N to U(1) which lives

on the D2-brane separated from the NS5-branes. The U(1) gauge field on this D2

52

would dualize to a compact scalar, parametrizing the x9 circle direction probed by

the D2-brane. More precisely, at k = 1, the vanishing condition of the potential

energy is given by

Φ(i)A a

(i)

αβ− a

(i−1)

αβΦ(i)A = 0 , ζ

(i)I + (σI)

ABΦ

(i)A ΦB(i) = (σI)

ABΦ

B(i−1)Φ(i−1)A . (2.27)

In the Higgs branch, one sets all a(i)

αβ’s to be equal, so that the first equation

is solved by breaking U(1)N → U(1). There is always a nonzero solution to the

next equations, meaning that the Higgs branch is always attached to the Coulomb

branch.

Since the Higgs branch now represents the strings leaving the NS5-branes, I

am only interested in the Coulomb branch CFT in the IR limit. However, the Higgs

branch cannot be detached from the Coulomb branch CFT by any deformation of

the theory. This is in contrast to the 2d gauge theories for the type IIB strings,

in which case the Higgs branch CFT of our interest could be detached from the

Coulomb branch CFT by turning on U(k) FI parameters. In fact, with generic FI

term ξ(i)I , the Coulomb branch will be all lifted by U(1)N → U(1). Since the elliptic

genus formula of [36, 37] is computing the index of CFT with generic nonzero FI

parameters, this formula will compute the unwanted Higgs branch index, with

lifted Coulomb branch. Apart from the absence of the SU(2)L2 in UV, this is

another reason that the above (4, 4) CFT is inconvenient for studying the little

string spectrum.

One can also add fractional D2-branes to this construction. Namely, the num-

ber of i’th D2-branes between i’th and i+1’th NS5-branes can be all different, ni,

forming a circular U(n1)× · · · × U(nN ) quiver.

2.3.1 N = (0, 4) gauge theory descriptions

As explained, the N = (4, 4) gauge theories for IIA little strings only see SU(2)D ⊂

SO(4) part of the R-symmetry. Although I expect the symmetry enhancement to

happen in IR, this means that the UV gauge theory would be of limited use.

53

x0 x1 x2 x3 x4 x5 x6 x7 x8 x11(S1)

N NS5 × × × × × × αini D2 × × (αi, αi+1)

1 D6 × × × × × × ×

Table 5: Brane construction of 2d N = (0, 4) gauge theory

Also, studying the spectrum of the Coulomb branch CFT will be difficult with the

approaches of [36, 37]. Closely following the idea of [40, 41], I shall engineer (0, 4)

UV gauge theories for the IIA string systems which resolve all these problems.

Now on top of the IIA branes explained after the x9-x11 flip, I also put one

D6-brane extended along 012345, 11 and localized at x6 = x7 = x8 = 0. See Table

5. Now with a D6-brane uplifting to the Taub-NUT space in M-theory, the gauge

theory SU(2) which rotates 678 directions in weakly coupled type IIA is interpreted

differently in the IR CFT of this gauge theory. Namely, the low energy limit of the

2d gauge theory is realized by taking the M-theory limit R′M → ∞ (after the 9-11

flip): see (2.26). So the embedding of the UV gauge theory’s symmetries into the

infrared R-symmetry has to be understood in the R′M → ∞ limit, where I have R4.

The SO(3) rotating the asymptotic R3 of Taub-NUT rotates the R4 as SU(2)R2 in

‘IR.’ Also, after compactifying one more circle x5, I can turn on a background gauge

field of the D6-branes, as Ai5 + iAi11 ≡ mi ∼ (m, 2m, 3m, · · · , Nm) with nonzero

B5,11 turned on. The parameter m realizes the chemical potential for the Cartan of

SU(2)L2 [40, 41]. Thus, we can turn on full set of SO(4)R chemical potentials of the

partition function in this setting. From the 2d gauge theory viewpoint, adding one

D6-brane just affects the way I connect the UV regime R3 × S1 at weak-coupling

with the IR regime R4 at strong coupling. Since the IR brane configuration is

complete the same as the original M2-M5 system, I expect the (0, 4) gauge theory

to flow to the same (4, 4) CFT on the Coulomb branch. (However, see section 4 for

discussions on irrelevant decoupled sectors within this gauge theory.)

A 2d N = (0, 4) UV gauge theory is engineered from this brane setting, with

54

U(nN)

U(1)N

U(1)1 U(1)2

U(1)3

U(1)4

U(n2)U(n1)

U(n3)

U(n4)

Figure 3: AN−1 quiver diagram of the 2d N = (0, 4) gauge theory for the IIAstrings. Solid lines denote the hypermultiplets, thin dashed lines denote the Fermimultiplets, and thick dashed lines denote the twisted hyper multiplets.

Multiplet Fields U(ni) U(1)m

Vector A(i)µ , λ

(i)Aα+ adji 0

Hyper q(i)α , ψ

(i)A− ni 0

Hyper a(i)

αβ, λ

(i)Aα− adji 0

Twisted hyper Φ(i)A , Ψ

(i)α− (ni−1, ni) 1

Fermi Ψ(i)β+ (ni−1, ni) 1

Fermi ψ(i)+ ni 1

Fermi ψ(i)+ ni −1

Table 6: Fields of the N = (0, 4) quiver gauge theory

55

supercharges given by QAα. The fields can be characterized again by a circular

quiver of Fig. 3. Each circular node involves N = (0, 4) U(ni) gauge multipletiplet

(Aµ, λAα+ ), and a N = (0, 4) adjoint hypermultiplet (aαβ, λ

Aα−), denoted by the

solid lines. ni’s are the number of the D2-branes suspended between adjacent NS5-

branes. Thick dashed lines between two circular nodes denote the bi-fundamental

twisted hypermultiplets (ΦA,Ψα−). Thin dashed lines between two circular nodes

denote the bi-fundamental fermi multiplets Ψβ+. D6-brane introduces extra fields:

fundamental hyper multiplets (qα, ψA−) and Fermi multiplets ψ+, ψ+. These fields

are are summarized in Table. 6. As explained in the previous paragraph, and just

like [40], the chemical potentials for U(1)i and U(1)i+1 are locked asmi+1−mi = m,

so that one just has one U(1)m. Compared to the previous (4, 4) gauge theory for

the IIA strings, the (0, 4) fields on the first and third lines of Table. 6 are forming

the (4, 4) vector multiplet, which I decomposed as above since the system does not

preserve (4, 4) SUSY. Also, the fields on the fourth and fifth lines form the previous

(4, 4) hypermultiplet. They again make a twisted Higgs branch, which represents

the degrees of freedom of fully winding D2-branes leaving the NS5-branes. The

Coulomb branch of the (4, 4) theory is replaced here by the Higgs branch formed

by the second and third lines, which is our main interest to study the IIA little

strings.

The SUSY action of the (0, 4) gauge theory can also be easily constructed.

From the (0, 2) supersymmetric formalism, one has to determine the holomorphic

potentials EΨ, JΨ for each Fermi multiplet Ψ, ensuring the (0, 4) SUSY enhance-

ment. For instance, see [42] for how this can be done. Here, following [42], I simply

write down these potentials for our theory. Let us call the (0, 2) Fermi multiplet

from the (0, 4) vector multiplet as Λi, which is made of λ11 and λ22. Then one

should first take

JΛi = qiqi + [Bi, Bi]− ξC , EΛi = Φi+1Φi+1 − ΦiΦi , (2.28)

for (0, 4) SUSY [42]. Here and below, I use the chiral superfield notation qα = (q, q†),

56

a1β = (B, B†), ΦA = (Φ, Φ†) for a while. I also inserted the FI parameter ξC for later

use, which corresponds to turning on worldvolume Bµν field on 1234 directions.

The above J,E should be accompanied by other J,E functions for other Fermi

fields, to satisfy∑

ΨEΨJΨ = 0 after summing over all Fermi multiplets Ψ. This is

another requirement from SUSY. To meet the last condition, one should turn on

the following potentials for other Fermi multiplet fields:

Eψi= qi−1Φi , Jψi

= −Φiqi−1 , Eψi= Φi+1qi+1 , Jψi

= qi+1Φi+1

EΨi = ΦiBi −Bi−1Φi , JΨi = BiΦi − ΦiBi−1 ,

EΨi= Bi−1Φi − ΦiBi , JΨi

= BiΦi − ΦiBi−1 . (2.29)

The bosonic potential is V =∑

Ψ(|JΨ|2 + |EΨ|2) + 12

∑iD

2i with Di given by

Di = qiq†i − q†i qi + [Bi, B

†i ] + [Bi, B

†i ]−Φ†

iΦi + ΦiΦ†i +Φi+1Φ

†i+1 − Φ†

i+1Φi+1 − ξR .

(2.30)

After some rearrangement, one obtains

V =1

2tr

N∑i=1

(qiα(σ

m)αβqβi +

1

2(σm)α

β[aiαα, a

αβi ]− ξm

)2

+ tr

N∑i=1

((σI)ABΦiAΦ

Bi − (σI)ABΦ

Bi−1Φi−1,A

)2+ tr

N∑i=1

(|ΦiAqiα|2 + |Φ†

i+1,Aqiα|2 + |ΦiAaiαβ − ai−1,αβΦiA|

2)

(2.31)

where ξ3 = ξR and ξ1 + iξ2 ∼ ξC, with manifest SU(2)R1 × SU(2)R2 symmetry.

Note that with nonzero ξm, qiα fields are required to be nonzero at low energy,

which lift the twisted Higgs branch of ΦiA. Namely, even if n1 = n2 = · · · =

nN , they cannot combine and leave the NS5-branes unlike the N = (4, 4) model.

Also, the previous (4, 4) Coulomb branch fields aαβ form (0, 4) Higgs branch fields,

together with new degrees qα. The (0, 4) setting will thus be computing the correct

little string elliptic genus. However, the (0, 4) elliptic genus will also capture a subtle

57

trace of the presence of a D6-brane from the sector with n1 = n2 = · · · = nN , in

which case the D2-branes make full windings along x11. This can be easily accounted

for and factored out, after which I shall be obtaining the IIA little string index. I

shall explain this in section 4.

2.3.2 The elliptic genus of IIA little strings

I define the index of IIA little string theory wrapping a spatial circle along x5, as

follows,

ZIIA(αi, ϵ±,m; q′, w′)

= Tr[(−1)F q′HL q′HRw′ke2πiαiΠie2πiϵ−(2JL1)e2πim(2J2L)e2πiϵ+(2J1R+2J2R)

]. (2.32)

Πi are charges of the self-dual tensor fields, supported on each M5-brane, with

the chemical potentials, αi. q′ are the fugacity variable counting the number of

momentum, and w′ the winding fugacity of the IIA little strings. 2παiRM are the

positions of the 5-branes along x11.

An M2-brane suspended between the i’th interval between the M5-branes,

(αi, αi+1), carries nonzero charges Πi = 1 and Πi+1 = −1. The charges ei − ei+1

form the simple roots of the AN−1 algebra, for i = 1, · · · , N − 1. The last charge

eN − e1, is accompanied by an extra winding, so the whole N roots become the

simples roots of AN−1. The fugacity variables corresponding to these simple roots

are given by

v1 ≡ e2πiα12 , v2 ≡ e2πiα23 , · · · , vN−1 ≡ e2πiαN−1,N ,

vN ≡ e2πiαN,N+1 = e2πiαN,1w′ . (2.33)

where αij = αi − αj . For convenience, I introduce αN+1, where e−2πiαN+1 =

e−2πiα1w′.

For large RA, the low energy degrees living on the winding strings decouple

from the 6d degrees on the 5-branes. So the index of the IIA little string theory on

58

R4 × T 2 factorizes as

ZIIA(αi, ϵ±,m; q′, w′) = ZIIAmom(ϵ±,m; q′)ZIIA

string(αi, ϵ±,m; q′, w′) . (2.34)

ZIIAmom comes from the momenta on N separated M5-branes wrapping a spatial

circle. Unlike the IIB perturbative index ZIIBpert which had massive W-boson contri-

butions, the IIA 5-brane does not have extra massive particle states in it (because it

only has strings). So this contribution should factorize into N single 5-brane contri-

butions. It can be computed either from N Abelian tensor multiplet, or equivalently

from the multiple D0-brane index bound to a single D4-brane [32]. The result is

ZIIAmom(ϵ±,m; q′) = PE

[NI−(ϵ1,2,m)

q′

1− q′

], (2.35)

where

I−(ϵ1,2,m) ≡sinh 2πi(m+ϵ−)

2 sinh 2πi(m−ϵ−)2

sinh 2πiϵ12 sinh 2πiϵ2

2

(2.36)

with ϵ± = ϵ1±ϵ22 .

The contribution ZIIAstring comes from the elliptic genera of the (0, 4) gauge

theory theory that I have explained in the previous subsection. This elliptic genus

can be computed by the contour integral using the Jeffrey-Kirwan residues [36, 37],

or the refined topological vertex method with (p, q)-fivebrane web obtained by T-

dualizing the branes along x5 [40]. By summing up the elliptic genera over all

possible ni numbers, one obtains ZIIAstring. The result is labeled by sets of N Young

diagrams, Y1, ..., YN, where |Yi| = ni,

ZIIAstring(αi, ϵ±,m; q′, w′) =

∞∑ni=0

e2πi∑N

i=1 niαi,i+1Z(n1,...,nN )string (ϵ±,m; q′)

=∞∑ni=0

(v1)n1(v2)

n2 · · · (vN )nNZ(n1,...,nN )string (ϵ±,m; q′). (2.37)

59

Z(n1,...,nN )string (ϵ±,m; q′) is the elliptic genus with fixed ni’s, which is given by

Z(n1,...,nN )string (ϵ±,m; q′)

=∑

Y1,··· ,YN;|Yi|=ni

N∏i=1

∏(a,b)∈Yi

θ1(q′;E

(a,b)i,i+1 −m+ ϵ−)θ1(q

′;E(a,b)i,i−1 +m+ ϵ−)

θ1(q′;E(a,b)i,i + ϵ1)θ1(q′;E

(a,b)i,i − ϵ2)

,

(2.38)

where

E(a,b)ij = (Yi,a − b)ϵ1 − (Y T

j,b − a)ϵ2 , E(a,b)i,N+1 = E

(a,b)i,1 . (2.39)

(a, b) denotes the position of each box in a Young diagram. Ya,i is the length of the

a’th row of the Young diagram Yi. YTa,i is the length of the a’th column of Yi.

As I emphasized earlier in this section, the contribution from the (4, 4) Higgs

branch (or the (0, 4) twisted Higgs branch) formed by Φ(i)A is not completely de-

coupled. I shall explain at the beginning of section 4 what contribution I expect to

get from this decoupled sector.

2.4 T-duality of protected little string spectra

The IIB little string theory on a circle is supposed to be T-dual to IIA little string

theory on the dual circle, with the radiii related by RA = α′

RB. The winding IIB

little strings on S1B is dual to the momentum on S1

A, and vice versa. Their BPS

masses agree with each other, since

mIIB winding =2πRB

2πα′ =RB

α′ =1

RA= mIIA momentum . (2.40)

The fractional momenta of IIB little string theory are dual to the fractional winding

numbers of IIA little strings,

mIIB momentum =αi,i+1

RB

T−dual−−−−−→ (αi,i+1)RA

α′ = αi,i+1(2πRA)TF1 ,

60

T-duality

x5x6789

x01234

NS5

x01234

x11

x5

M5 M5 M5

M2 M2 M2 M2

Figure 4: T-duality and M-theory uplift of the IIB setting. The solid lines representthe winding little strings. The dashed lines represent the momentum along thecircle.

where αij = αi − αj . T-duality between two little string theories is demonstrated

by Fig. 4.

T-duality between IIA and IIB little string theories would naively imply

ZIIA(αi, ϵ±,m; q′, w′)|q′→w,w′→q = ZIIB(αi, ϵ±,m; q, w) . (2.41)

As I stated at the end of section 3.1, using an alternative 2d (0, 4) gauge theory

to compute the IIA elliptic genus (and thus the IIA little string index) will leave

a subtle trace of the fact that I made a UV deformation of the gauge theory by

putting an extra D6-brane. A spectrum change will happen in a sector with full

wound D2-branes along x11, namely with states carrying the factors of fugacities

w′ but not αi’s. Let us first explain this small subtlety in our IIA calculation.

On the IIB side, consider the single particle states with zero electric charges

(no dependence on αi) and zero winding (w0 order). This contribution is contained

in the ZIIBpert factor. In particular, it comes from the N Cartan modes of the 6d

U(N) SYM. Their partition function is given by

PE

[NI+(ϵ1,2,m)

q

1− q

]. (2.42)

61

In the IIA side, this will correspond to a sector with fully wound D2-branes along

x11, at (q′)0 order with n1 = n2 = · · · = nN . At ξm = 0 in section 3.1, there is

an extra twisted Higgs branch which meets the Higgs branch of our interest. The

former sector will represent the little strings leaving NS5-branes. The two sectors

would decouple in IR, but the 2d gauge theory contains both in its Hilbert space.

Now by turning on the FI term ξm, the continuum of twisted Higgs branch will be

lifted. However, after turning on ξm, it often happens that there appear extra bound

states of the continuum degrees with the remaining 2d strings of our interest. For

instance, see [38] and references therein for many occasions in which extra bound

states occur at nonzero FI parameters. From the (0, 4) computation of the IIA side,

I shall find

PE

[I−

w′

1− w′ + (N − 1)I+w′

1− w′

]= PE

[NI+(ϵ1,2,m)

w′

1− w′

]· Zextra(w

′)

(2.43)

in the same sector, instead of (2.42), where Zextra(w′) ≡

∏∞n=1

11−(w′)n ∼ η(w′)−1.

I shall give an account for why Zextra should be appearing due to our (0, 4) defor-

mation of the UV theory. With this understood, I should define the true IIA index

as the expression computed in section 3.2 divided by Zextra. I call this

ZIIA(αi, ϵ1,2,m, q′, w′) =

ZIIA(αi, ϵ1,2,m, q′, w′)

Zextra(w′). (2.44)

I shall find that

ZIIA(αi, ϵ±,m; q′, w′)|q′→w,w′→q = ZIIB(αi, ϵ±,m; q, w) , (2.45)

which I checked for the cases with N = 1, 2, 3. This will establish the T-duality of

the strong-coupling little string spectra via the elliptic genus calculus.

I first explain (or at least heuristically understand) how Zextra would be ap-

pearing in our (0, 4) calculus. Consider the sector with n1 = · · · = nN ≡ n, forming

n full winding branes which have the right quantum number to leave the NS5-

62

branes. I weight n windings by (w′)n, and relax the constraint on fixed n. I would

like to count the BPS bounds of these strings with N NS5-branes directly, not using

the elliptic genus formula of [36, 37]. I shall do so without and with one D6-brane,

to clearly compare. For convenience, I T-dualize along the x11 circle, and obtain

many D1-branes along 05, N -centered Taub-NUT on 678 and 11 circle, and option-

ally a D5-brane along 012345 with Bµν (FI term) on 1234. Firstly, without D5, any

number n of wrapped D1-branes can form a bound state of multiply wound single

string. For each massive particle of this sort, I study its ground state wavefunction

on the N -centered Taub-NUT. This space has N normalizable harmonic forms, so

that there could be N possible bound states of the original N NS5-branes with this

particle. The index for this particle is thus NI+(ϵ1,2,m)(w′)n, where N comes from

N normalizable harmonic forms. The I+ factor appears because this is exactly the

same type of bound states as the half-BPS W-bosons in SYM, as in the IIB set-

ting. Summing over n and considering the multi-particle Hilbert space, one exactly

obtains (2.42) with q replaced by w′. Note that I have arrived at this conclusion

by a direct counting, without any deformation by continuous parameters, so this

should be part of the IIA little string index.

Now I consider the same problem after placing one D5-brane with FI parameter

(Bµν background). The setting of section 3 was that D6 and N NS5-branes are

placed at the same point of R3 in the 678 directions. Now T-dualizing along x11, one

finds a D5-brane on top of the R4/ZN singularity of unresolved Taub-NUT. Now,

among theN normalizable harmonic forms ofN -centered Taub-NUT,N−1 of them

are supported at the ZN singularity, where D5 is sitting. Since the fully winding

D1-branes are forced to be bound to D5 at the tip due to the FI parameter, D1-

branes stuck to D5 can still assume one of these N − 1 bound state wavefunctions.

The multi-particle index of the bounds is PE[(N − 1)I+w′

1−w′ ]. However, the last

normalizable harmonic form of Taub-NUT is not localized at the tip, so D1-branes

confined to D5 cannot be in this bound state. (The forbidden wavefunction is in

the twisted Higgs branch.) This accounts for the second term on the left hand

side of (2.43). Now, note that n D1-branes can also form threshold bounds with

63

single D5-brane, whose partition function is given by PE[I−w′

1−w′ ] [32]. (This extra

contribution is also from the twisted Higgs branch, since the D2-D6 bounds still

exist after displacing D6-NS5’s.) This explains the first term of (2.43), and thus the

origin of Zextra. By the discussions of this paragraph, it clearly comes from having

D6-brane and nonzero FI parameter, causing extra bound states or destroying some

in the twisted Higgs branch. So with this understood, T-duality would imply (2.45).

2.4.1 One NS5-brane

I start by considering the index of the U(1) IIB theory, although this should be a

free theory. The perturbative contribution is given by

ZIIBpert(ϵ±,m; q) = PE

[I+(ϵ±,m)

q

1− q

]. (2.46)

The U(1) instanton string partition function is given by

ZIIBstring(ϵ±,m; q, w) =

∑k=0

wkZk(ϵ±,m; q) , (2.47)

where

Zk =∑

Y :|Y |=k

∏s∈Y

θ1 (q;E(s) +m− ϵ−) θ1 (q;E(s)−m− ϵ−)

θ1 (q;E(s)− ϵ1) θ1 (q;E(s) + ϵ2), (2.48)

with

E(s) = −ϵ1h(s) + ϵ2v(s) . (2.49)

The full index of the U(1) theory is given by

ZIIB(ϵ±,m; q, w) = ZIIBpert(ϵ±,m; q)ZIIB

inst(ϵ±,m; q, w) (2.50)

To further explain this index, consider the single instanton string index given by

Z1(ϵ±,m; q) =θ1 (q;m± ϵ−)

θ1 (q; ϵ1) θ1 (q; ϵ2). (2.51)

64

where θ1(q; a ± b) ≡ θ1(q; a + b)θ1(q; a − b). In terms of Z1, I find that the multi-

instanton string index is given by the Hecke transformation of Z1,

Zinst(ϵ±,m; q, w) = exp

∞∑n=1

1

nwn∑ad=na,d∈Z

∑b(mod d)

Z1

(aϵ±, am;

aτ + b

d

) (2.52)

where q = e2πiτ . This is checked up to high orders in w and q.

The partition function given by the Hecke transformation appears, for instance,

in conformal field theories on symmetric product target spaces. This is closely

related to the fact that the moduli-space of U(1) multi-instantons is a symmetric

product of the single instanton moduli space R4. More precisely, the symmetric

product CFT was suggested to be the theory at nonzero world-sheet theta angle

θ = π [25, 29]. Since the elliptic genus would be insensitive to the continuous

parameters, away from ζI = 0, θ = 0, it is natural to have (2.52).

On the IIA side, the 2d N = (0, 4) quiver gauge theory itself has an enhanced

N = (4, 4) SUSY, and becomes precisely the same to the 2d N = (4, 4) ADHM

gauge theory for IIB strings. Therefore,

ZIIAstring(ϵ±,m; q′, w′) = ZIIB

inst(ϵ±,m, q′, w′) . (2.53)

The extra factor ZIIAmom on the IIA side is given by

ZIIAmom(ϵ±,m; q′) = PE

[I−(ϵ±,m)

q′

1− q′

]= ZIIB

pert(ϵ±,m, q′)Zextra(q

′) . (2.54)

So the T-duality relation (2.45) is equivalent to ZIIA(ϵ±,m; q′, w′) being invariant

under the exchange of q′ and w′. The last property is in fact true, which can be

understood as the geometric duality of the 5-brane web obtained by T-dualizing

our IIA brane setting along x5 [43], as shown in Fig. 5. If I write ZIIA(ϵ±,m; q′, w′)

as

ZIIA(ϵ±,m; q′, w′) = PE[I−(ϵ±,m)zsp(ϵ±,m, q

′, w′)], (2.55)

65

a) b)

NS5

D5

NS5

D5

D5

NS5

Figure 5: a) (p, q) fivebranes web dual to rank 1 little string theory. b) Trialitybetween three Kahler parameters, q = qy−1, w = wy−1, and y.

zsp(ϵ±,m) is given by

zsp(ϵ±,m; q′, w′) = (q′ + w′) + (q′2 + w′2) + (q′w′)

[tu+

t

u+

1

tu+u

t− uy

−yu− u

y− 1

uy

]+ q′3 + w′3 + (q′2w′ + q′w′2)

[t2u2 +

t2

u2+u2

t2+

1

t2u2+ t2

+1

t2− tu2y − ty

u2− tu2

y− t

u2y− y

tu2− u2

ty− 1

tu2y− u2y

t+ tu+

t

u+

1

tu

+u

t− 2ty − 2t

y− 2

ty− 2y

t+ 2u2 +

2

u2− uy − y

u− u

y− 1

uy+ y2 +

1

y2+ 4

]+ (q′4 + w′4) + (q′3w′ + q′w′3)

[t3u3 +

t3

u3+u3

t3+

1

t3u3+ t3u+

t3

u+u

t3+

1

t3u

− t2u3y − t2y

u3− t2u3

y− t2

u3y− u3y

t2− y

t2u3− u3

t2y− 1

t2u3y+ t2u2 +

t2

u2+u2

t2

+1

t2u2− 2t2uy − 2t2y

u− 2t2u

y− 2t2

uy− 2uy

t2− 2y

t2u− 2u

t2y− 2

t2uy+ 2t2 +

2

t2

+ 2tu3 +2t

u3+

2

tu3+

2u3

t− 2tu2y − 2ty

u2− 2tu2

y− 2t

u2y− 2y

tu2− 2u2

ty− 2

tu2y

− 2u2y

t+ tuy2 +

ty2

u+tu

y2+

t

uy2+y2

tu+

u

ty2+

1

tuy2+uy2

t+ 6tu+

6t

u+

6

tu

+6u

t− 4ty − 4t

y− 4

ty− 4y

t− u3y − y

u3− u3

y− 1

u3y+ u2y2 +

y2

u2+u2

y2+

1

u2y2

(2.56)

66

+4u2 +4

u2− 5uy − 5y

u− 5u

y− 5

uy+ 2y2 +

2

y2+ 8

]+ (q′2w′2)

[u4t4 + u2t4

+t4

u2+t4

u4+ t4 + u3t3 + 2ut3 − u4yt3 − 2u2yt3 − 2yt3 +

2t3

u− 2yt3

u2+t3

u3

− yt3

u4− u4t3

y− 2u2t3

y− 2t3

y− 2t3

u2y− t3

u4y+ 2u4t2 + 7u2t2 + u2y2t2 +

y2t2

u2

+ y2t2 − 2u3yt2 − 5uyt2 − 5yt2

u+

7t2

u2− 2yt2

u3+

2t2

u4− 2u3t2

y− 5ut2

y− 5t2

uy

− 2t2

u3y+u2t2

y2+t2

y2+

t2

u2y2+ 9t2 + 5u3t+ u3y2t+ 4uy2t+

4y2t

u+y2t

u3+ 15ut

− u4yt− 7u2yt− 12yt+15t

u− 7yt

u2+

5t

u3− yt

u4− u4t

y− 7u2t

y− 12t

y− 7t

u2y

− t

u4y+u3t

y2+

4ut

y2+

4t

uy2+

t

u3y2+

2u4

t2+u4

t4+ 2u4 +

u3

t3− uy3 +

7u2

t2+u2

t4

+ 12u2 +u2y2

t2+ 2u2y2 +

y2

t2+

4y2

tu+

2y2

u2+

y2

t2u2+

y2

tu3+ 5y2 +

2u

t3− 4u3y

− 14uy − 2u3y

t2− 5uy

t2+

9

t2− u4y

t3− 2u2y

t3− 2y

t3+

1

t4− y3

u− 14y

u+

15

tu− 5y

t2u

+2

t3u+

12

u2− 7y

tu2+

7

t2u2− 2y

t3u2+

1

t4u2− 4y

u3+

5

tu3− 2y

t2u3+

1

t3u3+

2

u4

− y

tu4+

2

t2u4− y

t3u4+

1

t4u4− 4u3

y− 14u

y− u4

ty− 7u2

ty− 12

ty− 2u3

t2y− 5u

t2y

− u4

t3y− 2u2

t3y− 2

t3y− 14

uy− 5

t2uy− 7

tu2y− 2

t3u2y− 4

u3y− 2

t2u3y− 1

tu4y

− 1

t3u4y+

2u2

y2+

5

y2+u3

ty2+

4u

ty2+

u2

t2y2+

1

t2y2+

4

tuy2+

2

u2y2+

1

t2u2y2

+1

tu3y2− u

y3− 1

uy3+ 22 +

5u3

t+u3y2

t+

4uy2

t+

15u

t− u4y

t− 7u2y

t

−12y

t

]+ · · · . (2.57)

where t = e2πiϵ+ , u = e2πiϵ− , y = e2πim. I checked the symmetry of q′ ↔ w′

exchange up to 5th orders in q′ and w′.

Furthermore, defining the following variables,

q = qy−1 , w = wy−1 . (2.58)

67

triality of exchanging (q, w, y) has been discovered in [43]. This is also a geometric

duality of Fig. 5. Triality is simply realized on the universal covering of the torus,

as a subgroup of Sp(4, Z) duality. To deal with (q, w, y) in equal footing, I redefine

the index, including extra perturbative contributions at y ≪ 1, as

Z(ϵ±; q, w, y) = PE [Icom(ϵ±)y]ZIIA . (2.59)

Icom(ϵ±) is given by

Icom(ϵ±) =1

2 sinh 2πiϵ12 2 sinh 2πiϵ2

2

=t

(1− tu)(1− tu−1). (2.60)

Writing Z as

Z(ϵ±; q, w, y) = PE[Icomzsp(ϵ±; q, w, y)

], (2.61)

zsp is given by

zsp(ϵ±; q, w, y) = q + w + y − (qw + qy + wy)(u+ u−1)

+ qwy(1 + u2)(t+ u+ t2u+ tu2)

tu2+ (q2w + qw2 + q2y + qy2 + w2y + wy2)

− (q2w2 + q2y2 + w2y2)(u+ u−1)− qwy(q + w + y)

(u2 + 1

) (t2(u2 + 1

)+ 2tu+ u2 + 1

)tu2

+ (q3w2 + q2w3 + q3y2 + q2y3 + w3y2 + w2y3)

+ qwy(q2 + w2 + y2)(1 + u2)(t+ u+ t2u+ tu2)

tu2

+ qwy(qw + qy + wy)t4(u5 + u3 + u

)+ t3

(u6 + 4u4 + 4u2 + 1

)t2u3

+ qwy(qw + qy + wy)t2(3u4 + 7u2 + 3

)u+ t

(u6 + 4u4 + 4u2 + 1

)+ u5 + u3 + u

t2u3

− (u+ u−1)(q3w3 + q3y3 + w3y3)

− qwy(q2w + qw2 + (cyclic))

(u2 + 1

) (t4(u4 + u2 + 1

)+ 3t3

(u3 + u

))t2u3

− qwy(q2w + qw2 + (cyclic))

(u2 + 1

) (2t2(u4 + 3u2 + 1

)+ 3t

(u3 + u

)+ u4 + u2 + 1

)t2u3

+ · · · (2.62)

68

reconfirming the expected triality of [43]. It is curious to note that the triality

implies the T-duality of IIA/IIB strings.

2.4.2 Two NS5-branes

The index of U(2) IIB little string theory is given by

ZIIB(αi, ϵ±,m; q, w) = ZIIBpert(αi, ϵ±,m; q)ZIIB

inst(αi, ϵ±,m; q, w) , (2.63)

where

ZIIBpert(αi, ϵ±,m; q) =PE

[I+v1 +

(2I+ + I+(v1 + v−1

1 )) q

1− q

]=PE

[I+v1 + v2 + 2v1v2

1− v1v2

], (2.64)

with v1 = e2πiα12 , and v2 ≡ qv−11 . I+ is given by eq.(2.20). ZIIB

inst is given by

ZIIBinst(αi, ϵ±,m; q, w) =

∞∑k=0

wkZk(αi, ϵ±,m; q) . (2.65)

Zk is obtained from eq.(2.24). Expanding ZIIBinst(αi, ϵ±,m; q, w) with w, v1, v2 =

qv−11 , one obtains

ZIIBinst(ϵ±,m;w, vi)

= 1− w2t(u− y)(uy − 1)

y(t− u)(tu− 1)+ w(v1 + v2)

(t2 + 1

)(t− y)(ty − 1)(y − u)(uy − 1)

ty2(t− u)(tu− 1)

+ w(v21 + v22)

(t2 + 1

) (t4 + 1

)(t− y)(ty − 1)(y − u)(uy − 1)

t3y2(t− u)(tu− 1)

+ wv1v22(t− y)(ty − 1)(y − u)(uy − 1)

(t2uy + t(u− y)(uy − 1) + uy

)tuy3(t− u)(tu− 1)

+ w(v21v2 + v1v22)

(t2 + 1

)(t− y)(ty − 1)(y − u)(uy − 1)

t3uy3(t− u)(tu− 1)

×− t(t+ u)(1 + tu)(1 + y2) + (t+ u+ t2u)(1 + t(t+ u))y

+ · · · (2.66)

69

The index for the rank 2 IIA little string theory is given by

ZIIA = Zextra(q)−1ZIIA

mom(ϵ±,m;w)ZIIAstring(αi, ϵ±,m;w, q) (2.67)

where I inserted q′ = w, w′ = q. Zextra(q) is given by

Zextra(q) = PE

[q

1− q

]= PE

[v1v2

1− v1v2

]. (2.68)

ZIIAmom(ϵ±,m;w) is given by

ZIIAN=2 mom(ϵ±,m;w) = PE

[2I−(ϵ±,m)

w

1− w

]. (2.69)

ZIIAstring(αi, ϵ±,m;w, q) takes the form of

ZIIAstring(αi, ϵ±,m;w, q) =

∞∑n1,n2=0

(v1)n1(v2)

n2Z(n1,n2)string (ϵ±,m;w) , (2.70)

Note that Z(n1,n2)string (ϵ±,m;w) = Z

(n2,n1)string (ϵ±,m;w), from the symmetry of the quiver.

Z(n1,n2)string (ϵ±,m;w) can be easily obtained from (2.38). For instance,

Z(1,0)string(ϵ±,m;w) =

θ1(w,m± ϵ+)

θ1(w, ϵ1)θ1(w, ϵ2), Z

(1,1)string(ϵ±,m;w) =

θ1(w,m± ϵ−)2

θ1(w, ϵ1)2θ1(w, ϵ2)2

(2.71)

Z(2,0)string(ϵ±,m;w) = Z

(1,0)string ·

(θ1 (w; ϵ+ + ϵ1 ±m)

θ1 (w; 2ϵ1) θ1 (w; ϵ1 − ϵ2)− (ϵ1 ↔ ϵ2)

)(2.72)

I write the indices of the IIA/IIB little string theories as

ZIIB(αi, ϵ±,m;w, vi) = PE

Icom(t, u) ∞∑i,j,k=0

F IIBijk (t, u, y)w

ivj1vk2

, (2.73)

ZIIA(αi, ϵ±,m;w, vi) = PE

Icom(t, u) ∞∑i,j,k=0

F IIAijk (t, u, y)w

ivj1vk2

, (2.74)

70

where Icom is given by eq.(2.60). The coefficients F IIBijk (t, u, y) are polynomials of

t = e2πiϵ+ , u = e2πiϵ− , and y = e2πim. It is easily checked that F IIBijk (t, u, y) =

F IIBikj (t, u, y).

T-duality implies that F IIAijk = F IIB

ijk ≡ Fijk. I check T-daulity by comparing

F IIAijk and F IIB

ijk . I checked the agreements for

F000 = 1 , F010 = −t− 1

t+ y +

1

y, F011 = −2t− 2

t+ 2y +

2

y(2.75)

F020 = 0 , F021 = −t− 1

t+ y +

1

y, F022 = −2t− 2

t+ 2y +

2

y(2.76)

F100 = −2u− 2

u+ 2y +

2

y, (2.77)

F110 = −t2u− t2

u− u

t2− 1

t2u+ t2y +

t2

y+y

t2+

1

t2y+ tuy +

ty

u+tu

y+

t

uy

+y

tu+u

ty+

1

tuy+uy

t− ty2 − t

y2− 1

ty2− y2

t− 2t− 2

t− 2u− 2

u

+ 2y +2

y(2.78)

F111 = −2t2u− 2u

t2− 2t2

u− 2

t2u+ 2t2y +

2y

t2+

2

t2y+

2t2

y− 2tu2 − 2u2

t− 2t

u2

− 2

tu2+ 6tuy +

6uy

t+

6ty

u+

6y

tu+

6u

ty+

6t

uy+

6

tuy+

6tu

y− 4ty2 − 4y2

t− 4t

y2

− 4

ty2− 12t− 12

t+ 2u2y +

2y

u2+

2

u2y+

2u2

y− 4uy2 − 4y2

u− 4u

y2− 4

uy2− 12u

− 12

u+ 2y3 +

2

y3+ 14y +

14

y(2.79)

71

F120 = −t4u− t4

u− u

t4− 1

t4u+ t4y +

t4

y+y

t4+

1

t4y+ t3uy +

t3y

u+t3u

y+t3

uy

+uy

t3+

y

t3u+

u

t3y+

1

t3uy− t3y2 − t3

y2− y2

t3− 1

t3y2− 2t3 − 2

t3− 2t2u− 2t2

u

− 2u

t2− 2

t2u+ 2t2y +

2t2

y+

2y

t2+

2

t2y+ tuy +

ty

u+tu

y+

t

uy+

y

tu+u

ty+

1

tuy

+uy

t− ty2 − t

y2− 1

ty2− y2

t− 2t− 2

t− 2u− 2

u+ 2y +

2

y(2.80)

F121 = −t4u+ yt4 − t4

u+t4

y− u2t3 − 2y2t3 + 3uyt3 +

3yt3

u− t3

u2+

3ut3

y+

3t3

uy

− 2t3

y2− 6t3 + y3t2 − 3uy2t2 − 11ut2 + 2u2yt2 +

2yt2

u2+ 12yt2 − 3y2t2

u− 11t2

u

+2u2t2

y+

12t2

y+

2t2

u2y− 3ut2

y2− 3t2

uy2+t2

y3+ uy3t+

y3t

u− 5u2t− u2y2t− 9y2t

+ 13uyt+13yt

u− y2t

u2− 5t

u2+

13ut

y+

13t

uy− u2t

y2− 9t

y2− t

u2y2+ut

y3+

t

uy3− 24t

+y3

t2+y3

tu+ 2y3 − 6uy2 − 20u+

2u2y

t2+ 4u2y +

3uy

t3+

12y

t2+y

t4+

13y

tu+

3y

t3u

+4y

u2+

2y

t2u2+ 22y − 3uy2

t2− 11u

t2− u2

t3− 2y2

t3− 6

t3− u

t4− 6y2

u− 20

u− 3y2

t2u

− 11

t2u− 1

t4u− y2

tu2− 5

tu2− 1

t3u2+

4u2

y+

22

y+

13u

ty+

2u2

t2y+

12

t2y+

3u

t3y+

1

t4y

+13

tuy+

3

t3uy+

4

u2y+

2

t2u2y− 6u

y2− u2

ty2− 9

ty2− 3u

t2y2− 2

t3y2− 6

uy2− 3

t2uy2

− 1

tu2y2+

2

y3+

u

ty3+

1

t2y3+

1

tuy3+uy3

t− 5u2

t− u2y2

t− 9y2

t

+13uy

t− 24

t(2.81)

72

F122 = −2ut4 + 2yt4 − 2t4

u+

2t4

y− 2u2t3 − 4y2t3 + 6uyt3 +

6yt3

u− 2t3

u2+

6ut3

y

+6t3

uy− 4t3

y2− 12t3 − 2u3t2 + 2y3t2 − 8uy2t2 − 32ut2 + 8u2yt2 +

8yt2

u2+ 32yt2

− 8y2t2

u− 32t2

u− 2t2

u3+

8u2t2

y+

32t2

y+

8t2

u2y− 8ut2

y2− 8t2

uy2+

2t2

y3+ 6uy3t

+6y3t

u− 24u2t− 8u2y2t− 36y2t+ 2u3yt+ 52uyt+

52yt

u+

2yt

u3− 8y2t

u2− 24t

u2

+2u3t

y+

52ut

y+

52t

uy+

2t

u3y− 8u2t

y2− 36t

y2− 8t

u2y2+

6ut

y3+

6t

uy3− 88t− 2uy4

− 4u3 + 2u2y3 +2y3

t2+

6y3

tu+

2y3

u2+ 14y3 − 32uy2 − 88u+

8u2y

t2+ 24u2y

+6uy

t3+

32y

t2+

2y

t4+

52y

tu+

6y

t3u+

24y

u2+

8y

t2u2+

2y

tu3+ 94y − 2u3

t2− 8uy2

t2

− 32u

t2− 2u2

t3− 4y2

t3− 12

t3− 2u

t4− 2y4

u− 32y2

u− 88

u− 8y2

t2u− 32

t2u− 2

t4u− 8y2

tu2

− 24

tu2− 2

t3u2− 4

u3− 2

t2u3+

24u2

y+

94

y+

2u3

ty+

52u

ty+

8u2

t2y+

32

t2y+

6u

t3y

+2

t4y+

52

tuy+

6

t3uy+

24

u2y+

8

t2u2y+

2

tu3y− 32u

y2− 8u2

ty2− 36

ty2− 8u

t2y2− 4

t3y2

− 32

uy2− 8

t2uy2− 8

tu2y2+

2u2

y3+

14

y3+

6u

ty3+

2

t2y3+

6

tuy3+

2

u2y3− 2u

y4− 2

uy4

+6uy3

t− 24u2

t− 8u2y2

t− 36y2

t+

2u3y

t+

52uy

t− 88

t(2.82)

F200 = −2u− 2

u+ 2y +

2

y, (2.83)

73

F210 = −t3u2 − t3

u2− u2

t3− 1

t3u2+ t3uy +

t3y

u+t3u

y+t3

uy+uy

t3+

y

t3u+

u

t3y

+1

t3uy− 2t3 − 2

t3+ t2u2y +

t2y

u2+t2u2

y+

t2

u2y+u2y

t2+

y

t2u2+u2

t2y+

1

t2u2y

− t2uy2 − t2y2

u− t2u

y2− t2

uy2− uy2

t2− y2

t2u− u

t2y2− 1

t2uy2− 3t2u− 3t2

u− 3u

t2

− 3

t2u+ 3t2y +

3t2

y+

3y

t2+

3

t2y− 2tu2 − 2t

u2− 2

tu2− 2u2

t+ 4tuy +

4ty

u+

4tu

y

+4t

uy+

4y

tut+

4u

ty+

4

tuy+

4uy

t− 2ty2 − 2t

y2− 2

ty2− 2y2

t− 8− 8

t+ u2y +

y

u2

+u2

y+

1

u2y− 2uy2 − 2y2

u− 2u

y2− 2

uy2− 6u− 6

u+ y3 +

1

y3+ 7y +

7

y(2.84)

F211 = −ty4 − uy4 − y4

t− y4

u+ 2t2y3 + 2u2y3 + 6tuy3 +

6uy3

t+

2y3

t2+

6ty3

u

+6y3

tu+

2y3

u2+ 14y3 − t3y2 − u3y2 − 9tu2y2 − 33ty2 − 9t2uy2 − 33uy2 − 9u2y2

t

− 33y2

t− 9uy2

t2− y2

t3− 9t2y2

u− 33y2

u− 9y2

t2u− 9ty2

u2− 9y2

tu2− y2

u3+ 4tu3y

+4u3y

t+ 28t2y + 10t2u2y +

10u2y

t2+ 28u2y + 4t3uy + 52tuy +

52uy

t+

4uy

t3

+28y

t2+

4t3y

u+

52ty

u+

52y

tu+

4y

t3u+

10t2y

u2+

28y

u2+

10y

t2u2+

4ty

u3+

4y

tu3+ 90y

− 8t3 − 3t2u3 − 8u3 − 3t3u2 − 29tu2 − 86t− 29t2u− 86u− 29u2

t− 86

t− 3u3

t2

− 29u

t2− 3u2

t3− 8

t3− 29t2

u− 86

u− 29

t2u− 3t3

u2− 29t

u2− 29

tu2− 3

t3u2− 3t2

u3− 8

u3

− 3

t2u3+

4u3

ty+

52u

ty+

10u2

t2y+

28

t2y+

4u

t3y+

4t3

uy+

52t

uy+

52

tuy+

4

t3uy+

10t2

u2y

+28

u2y+

10

t2u2y+

4t

u3y+

4

tu3y− t3

y2− u3

y2− 9tu2

y2− 33t

y2− 9t2u

y2− 33u

y2− 9u2

ty2

− 33

ty2− 9u

t2y2− 1

t3y2− 9t2

uy2− 33

uy2− 9

t2uy2− 9t

u2y2− 9

tu2y2− 1

u3y2+

2t2

y3

+2u2

y3+

6tu

y3+

14

y3+

6u

ty3+

2

t2y3+

6t

uy3+

6

tuy3+

2

u2y3− t

y4− u

y4− 1

ty4

− 1

uy4+

4tu3

y+

28t2

y+

10t2u2

y+

28u2

y+

4t3u

y+

52tu

y+

90

y, (2.85)

74

NS5

NS5

NS5

NS5

D5

D5

D5

Figure 6: (p, q) fivebranes web dual to rank 2 little string theory

and further up to F444(t, u, y).

Let us define the following variables,

w = wy−1 , v1 = v1y−1 , v2 = v2y

−1 (2.86)

I can check w ↔ y exchange symmetry of the index. This is an analog of the triality

exchanging (q, w, y) at N = 1. It can be understood as a geometric duality of the

dual (p, q)-fivebrane web diagram Fig.6 of the rank 2 little string theory. Namely,

let us define the index

Z = PE[2Icomy]ZIIA . (2.87)

I find that Z is invariant under the w ↔ y exchange, to some high orders in

fugacities.

75

2.4.3 Three NS5-branes

The index of U(3) IIB little string theory is given by

ZIIB(αi, ϵ±,m; q, w) = ZIIBpert(αi, ϵ±,m; q)ZIIB

inst(αi, ϵ±,m; q, w) (2.88)

with

ZIIBpert = PE

[I+(v1 + v2 + v1v2) + 3I+

q

1− q

]× PE

[I+(v1 + v−1

1 + v2 + v−12 + v1v2 + v−1

1 v−12

) q

1− q

]= PE

[I+v1 + v2 + v3 + v1v2 + v1v2 + v2v3 + 3v1v2v3

1− v1v2v3

]. (2.89)

where v1 = e2πiα12 , v2 = e2πiα23 , and v3 = qv−11 v−1

2 . Zk’s appearing in ZIIBinst are

obtained from eq.(2.24).

The index of the rank 3 IIA little string theory is given by

ZIIA = Zextra(q)−1ZIIA

mom(ϵ±,m;w)ZIIAstring(αi, ϵ±,m;w, q)

= Zextra(q)−1ZIIA

mom(ϵ±,m;w)∞∑ni=0

vn11 vn2

2 vn33 Z

(n1,n2,n3)string (ϵ±,m;w) (2.90)

Zextra(q) is given by

Zextra = PE

[q

1− q

]= PE

[v1v2v3

1− v1v2v3

](2.91)

ZIIAmom(ϵ±,m;w) is given by

ZIIAmom(ϵ±,m;w) = PE

[3I−(ϵ±,m)

w

1− w

]. (2.92)

Note that Z(n1,n2,n3)string is invariant under the permutation of n1, n2 and n3, from

the symmetry of the quiver. The elliptic genera of the IIA fractional little strings,

76

Z(n1,n2,n3)string , are obtained from eq.(2.38). For example,

Z(1,0,0)string (ϵ±,m;w) =

θ1(w,m± ϵ+)

θ1(w, ϵ1)θ1(w, ϵ2), (2.93)

Z(1,1,0)string (ϵ±,m;w) =

θ1 (w;m± ϵ−) θ1 (w;m± ϵ+)

θ1 (w; ϵ1) 2θ1 (w; ϵ2) 2(2.94)

Z(2,0,0)string (ϵ±,m;w) = Z

(1,0,0)string ·

(θ1 (w; ϵ+ + ϵ1 ±m)

θ1 (w; 2ϵ1) θ1 (w; ϵ1 − ϵ2)− (ϵ1 ↔ ϵ2)

)(2.95)

Z(1,1,1)string (ϵ±,m;w) =

θ1(w,m± ϵ−)3

θ1(w, ϵ1)3θ1(w, ϵ2)3(2.96)

I write the IIB/IIA indices as

ZIIB(αi, ϵ±,m;w, vi) = PE

Icom(t, u) ∞∑i,j,k,l=0

F IIBijkl(t, u, y)w

ivj1vk2v

l3

, (2.97)

ZIIA(αi, ϵ±,m;w, vi) = PE

Icom(t, u) ∞∑i,j,k,l=0

F IIAijkl(t, u, y)w

ivj1vk2v

l3

, (2.98)

where Icom is given by eq.(2.60). The coefficients Fijkl(t, u, y) are polynomials of t,

u, and y, satisfying Fijkl = Fi(jkl).

T-duality implies that F IIAijkl = F IIB

ijkl ≡ Fijkl. I checked this for higher orders of

the fugacity variables. Fijkl are given by

F0100 = −t− 1

t+ y+

1

y, F0200 = 0 , F0110 = −t− 1

t+ y+

1

y, F0210 = 0 (2.99)

F0111 = −3t− 3

t+ 3y +

3

y, F0220 = 0 , F0211 = −t− 1

t+ y +

1

y(2.100)

F0221 = −t− 1

t+ y +

1

y, F0222 = −3t− 3

t+ 3y +

3

y, F1000 = −3u− 3

u+ 3y +

3

y(2.101)

77

F1100 =− t2u− t2

u− u

t2− 1

t2u+ t2y +

t2

y+y

t2+

1

t2y+ tuy +

ty

u+tu

y+

t

uy

+y

tu+u

ty+

1

tuy+uy

t− ty2 − t

y2− 1

ty2− y2

t− 2t− 2

t− 2u− 2

u

+ 2y +2

y(2.102)

F1200 =− t4u− t4

u− u

t4− 1

t4u+ t4y +

t4

y+y

t4+

1

t4y+ t3uy +

t3y

u+t3u

y

+t3

uy+uy

t3+

y

t3u+

u

t3y+

1

t3uy− t3y2 − t3

y2− y2

t3− 1

t3y2− 2t3 − 2

t3

− 2t2u− 2t2

u− 2u

t2− 2

t2u+ 2t2y +

2t2

y+

2y

t2+

2

t2y+ tuy +

ty

u+tu

y

+t

uy+

y

tu+u

ty+

1

tuy+uy

t− ty2 − t

y2− 1

ty2− y2

t− 2t− 2

t− 2u

− 2

u+ 2y +

2

y(2.103)

F1110 =− 2t2u− 2u

t2− 2t2

u− 2

t2u+ 2t2y +

2y

t2+

2

t2y+

2t2

y+ 3tuy +

3uy

t

+3ty

u+

3y

tu+

3u

ty+

3t

uy+

3

tuy+

3tu

y− 3ty2 − 3y2

t− 3t

y2− 3

ty2− 6t

− 6

t− uy2 − y2

u− u

y2− 1

uy2− 6u− 6

u+ y3 +

1

y3+ 7y +

7

y(2.104)

F1210 =− t4u− t4

u− u

t4− 1

t4u+ t4y +

t4

y+y

t4+

1

t4y+ 2t3uy +

2t3y

u+

2t3u

y

+2t3

uy+

2uy

t3+

2y

t3u+

2u

t3y+

2

t3uy− 2t3y2 − 2t3

y2− 2y2

t3− 2

t3y2− 4t3

− 4

t3− t2uy2 − t2y2

u− t2u

y2− t2

uy2− uy2

t2− y2

t2u− u

t2y2− 1

t2uy2− 5t2u

(2.105)

78

− 5t2

u− 5u

t2− 5

t2u+ t2y3 +

t2

y3+y3

t2+

1

t2y3+ 6t2y +

6t2

y+

6y

t2+

6

t2y

+ 4tuy +4ty

u+

4tu

y+

4t

uy+

4y

tu+

4u

ty+

4

tuy+

4uy

t− 4ty2 − 4t

y2− 4

ty2

− 4y2

t− 8t− 8

t− uy2 − y2

u− u

y2− 1

uy2− 6u− 6

u+ y3 +

1

y3+ 7y +

7

y

(2.106)

F1111 =− 6t2u− 6u

t2− 6t2

u− 6

t2u+ 6t2y +

6y

t2+

6

t2y+

6t2

y− 3tu2 − 3u2

t

− 3t

u2− 3

tu2+ 15tuy +

15uy

t+

15ty

u+

15y

tu+

15u

ty+

15t

uy+

15

tuy

+15tu

y− 12ty2 − 12y2

t− 12t

y2− 12

ty2− 30t− 30

t+ 3u2y +

3y

u2+

3

u2y

+3u2

y− 9uy2 − 9y2

u− 9u

y2− 9

uy2− 30u− 30

u+ 6y3 +

6

y3+ 36y +

36

y

(2.107)

F1220 = −2t4u− 2t4

u− 2u

t4− 2

t4u+ 2t4y +

2t4

y+

2y

t4+

2

t4y+ 3t3uy +

3t3y

u

+3t3u

y+

3t3

uy+

3uy

t3+

3y

t3u+

3u

t3y+

3

t3uy− 3t3y2 − 3t3

y2− 3y2

t3− 3

t3y2

− 6t3 − 6

t3− t2uy2 − t2y2

u− t2u

y2− t2

uy2− uy2

t2− y2

t2u− u

t2y2− 1

t2uy2

− 8t2u− 8t2

u− 8u

t2− 8

t2u+ t2y3 +

t2

y3+y3

t2+

1

t2y3+ 9t2y +

9t2

y+

9y

t2

+9

t2y+ 7tuy +

7ty

u+

7tu

y+

7t

uy+

7y

tu+

7u

ty+

7

tuy+

7uy

t− 7ty2 − 7t

y2

− 7

ty2− 7y2

t− 14t− 14

t− 2uy2 − 2y2

u− 2u

y2− 2

uy2− 12u− 12

u+ 2y3

+2

y3+ 14y +

14

y(2.108)

79

F1211 = −2ut4 + 2yt4 − 2t4

u+

2t4

y− u2t3 − 5y2t3 + 6uyt3 +

6yt3

u− t3

u2+

6ut3

y

+6t3

uy− 5t3

y2− 12t3 + 4y3t2 − 6uy2t2 − 22ut2 + 2u2yt2 +

2yt2

u2+ 26yt2

− 6y2t2

u− 22t2

u+

2u2t2

y+

26t2

y+

2t2

u2y− 6ut2

y2− 6t2

uy2+

4t2

y3− y4t+ 2uy3t

+2y3t

u− 5u2t− u2y2t− 24y2t+ 27uyt+

27yt

u− y2t

u2− 5t

u2+

27ut

y+

27t

uy

− u2t

y2− 24t

y2− t

u2y2+

2ut

y3+

2t

uy3− t

y4− 52t+

4y3

t2+

2y3

tu+ 9y3 − 13uy2

− 42u+2u2y

t2+ 4u2y +

6uy

t3+

26y

t2+

2y

t4+

27y

tu+

6y

t3u+

4y

u2+

2y

t2u2+ 51y

− 6uy2

t2− 22u

t2− u2

t3− 5y2

t3− 12

t3− 2u

t4− 13y2

u− 42

u− 6y2

t2u− 22

t2u− 2

t4u

− y2

tu2− 5

tu2− 1

t3u2+

4u2

y+

51

y+

27u

ty+

2u2

t2y+

26

t2y+

6u

t3y+

2

t4y+

27

tuy

+6

t3uy+

4

u2y+

2

t2u2y− 13u

y2− u2

ty2− 24

ty2− 6u

t2y2− 5

t3y2− 13

uy2− 6

t2uy2

− 1

tu2y2+

9

y3+

2u

ty3+

4

t2y3+

2

tuy3− 1

ty4− y4

t+

2uy3

t− 5u2

t− u2y2

t

− 24y2

t+

27uy

t− 52

t(2.109)

F2000 = −3u− 3

u+ 3y +

3

y(2.110)

F2100 = −t3u2 − t3

u2− u2

t3− 1

t3u2+ t3uy +

t3y

u+t3u

y+t3

uy+uy

t3+

y

t3u+

u

t3y

+1

t3uy− 2t3 − 2

t3+ t2u2y +

t2y

u2+t2u2

y+

t2

u2y+u2y

t2+

y

t2u2+u2

t2y

+1

t2u2y− t2uy2 − t2y2

u− t2u

y2− t2

uy2− uy2

t2− y2

t2u− u

t2y2− 1

t2uy2(2.111)

80

− 3t2u− 3t2

u− 3u

t2− 3

t2u+ 3t2y +

3t2

y+

3y

t2+

3

t2y− 2tu2 − 2t

u2− 2

tu2

− 2u2

t+ 4tuy +

4ty

u+

4tu

y+

4t

uy+

4y

tu+

4u

ty+

4

tuy+

4uy

t− 2ty2 − 2t

y2

− 2

ty2− 2y2

t− 8t− 8

t+ u2y +

y

u2+u2

y+

1

u2y− 2uy2 − 2y2

u− 2u

y2− 2

uy2

− 6u− 6

u+ y3 +

1

y3+ 7y +

7

y(2.112)

F2110 = −ty4 − y4

t+ 2t2y3 + 3tuy3 +

3uy3

t+

2y3

t2+

3ty3

u+

3y3

tu+ 7y3 − t3y2

− 2tu2y2 − 17ty2 − 7t2uy2 − 16uy2 − 2u2y2

t− 17y2

t− 7uy2

t2− y2

t3− 7t2y2

u

− 16y2

u− 7y2

t2u− 2ty2

u2− 2y2

tu2+ 19t2y + 5t2u2y +

5u2y

t2+ 9u2y + 4t3uy

+ 26tuy +26uy

t+

4uy

t3+

19y

t2+

4t3y

u+

26ty

u+

26y

tu+

4y

t3u+

5t2y

u2+

9y

u2

+5y

t2u2+ 45y − 8t3 − 3t3u2 − 12tu2 − 48t− 17t2u− 38u− 12u2

t− 48

t

− 17u

t2− 3u2

t3− 8

t3− 17t2

u− 38

u− 17

t2u− 3t3

u2− 12t

u2− 12

tu2− 3

t3u2+

26u

ty

+5u2

t2y+

19

t2y+

4u

t3y+

4t3

uy+

26t

uy+

26

tuy+

4

t3uy+

5t2

u2y+

9

u2y+

5

t2u2y

− t3

y2− 2tu2

y2− 17t

y2− 7t2u

y2− 16u

y2− 2u2

ty2− 17

ty2− 7u

t2y2− 1

t3y2− 7t2

uy2

− 16

uy2− 7

t2uy2− 2t

u2y2− 2

tu2y2+

2t2

y3+

3tu

y3+

7

y3+

3u

ty3+

2

t2y3+

3t

uy3

+3

tuy3− t

y4− 1

ty4+

19t2

y+

5t2u2

y+

9u2

y+

4t3u

y+

26tu

y+

45

y

(2.113)

and so on.

For the rank 3 indices, I can also check the duality exchanging w = wy−1 and

y.

81

2.5 SL(2, Z) transformations of the elliptic genus

The index of the winding IIB little strings is given by

ZIIBinst(αi, ϵ±,m; q, w)

=∞∑k=1

wk∑

Y :∑

i |Yi|=k

N∏i,j=1

∏s∈Yi

θ1 (q;Eij +m− ϵ−) θ1 (q;Eij −m− ϵ−)

θ1 (q;Eij − ϵ1) θ1 (q;Eij + ϵ2)(2.114)

where

Eij = αi − αj − ϵ1hi(s) + ϵ2vj(s) . (2.115)

q and w are given by

q = e2πiτ , w = e2πiρ , (2.116)

where τ = iRβ

RIIBis the complex structure on the torus, α′ρ = iRβRIIB is the Kahler

parameter of it, and Rβ is the radius of the temporal circle.

The modular transformation of the Jacobi’s theta function is given by

θ1

(−1

τ;z

τ

)= −i(−iτ

12 ) exp

(iπz2

τ

)θ1(τ ; z) (2.117)

Using this property, the S-duality transformation of ZIIBinst in τ is given by

∞∑k=1

wk exp

(−2πi

m2 − ϵ2+τ

kN

)

×∑

Y :∑

i |Yi|=k

N∏i,j=1

∏s∈Yi

θ1

(− 1τ ;

Eij+m−ϵ−τ

)θ1

(− 1τ ;

Eij−m−ϵ−τ

)θ1

(− 1τ ;

Eij−ϵ1τ

)θ1

(− 1τ ;

Eij+ϵ2τ

) . (2.118)

Transforming the fugacity variable for the winding number, w by

w → w = e−2πim2−ϵ2+

τNw , (2.119)

82

ZIIBinst is invariant under the following transformation,

q = e2πiτ → q = e−2πiτ , w → w = e−2πi

m2−ϵ2+τ

Nw . (2.120)

The elliptic genus of the IIA strings is given by

ZIIAstring(αi, ϵ±,m; q′, w′)

=∞∑ni=0

e2πi∑N

i=1 niαi,i+1Z(n1,...,nN )string (ϵ±,m; q′)

=

∞∑ni=0

(v1)n1 · · · (vN )nNZ

(n1,...,nN )string (ϵ±,m; q′)

=∞∑ni=0

(v1)n1−nN · · · (vN−1)

nN−1−nN (w′)nNZ(n1,...,nN )string (ϵ±,m; q′) (2.121)

where vi = e2πi(αi,i+1) and e−2πi(αN+1) = e−2πiα1w′. Z(n1,...,nN )string is given by

Z(n1,...,nN )string (ϵ±,m; q′)

=∑

Y1,··· ,YN;|Yi|=ni

N∏i=1

∏(a,b)∈Yi

θ1(q′;E

(a,b)i,i+1 −m+ ϵ−)θ1(q

′;E(a,b)i,i−1 +m+ ϵ−)

θ1(q′;E(a,b)i,i + ϵ1)θ1(q′;E

(a,b)i,i − ϵ2)

,

(2.122)

where

E(a,b)ij = (Yi,a − b)ϵ1 − (Y T

j,b − a)ϵ2 . (2.123)

Upon T-duality transformation, the complex structure and the Kahler parameter

are exchanged. The modular transformation of Z(n1,...,nN )string in ρ ≡ τ ′ is given by

Z(n1,··· ,nN )string (ϵ±,m; ρ) = exp

[−πiρ

(ϵ1ϵ2

N∑a=1

(na − na+1)2 + 2(m2 − ϵ2+)

N∑a=1

na

)]

× Z(n1,··· ,nN )string

(ϵ±ρ,m

ρ;−1

ρ

). (2.124)

83

where nN+1 = n1. Via T-duality relation, this would imply a definite S-duality

transformation of ZIIB in ρ, which would have been difficult to obtain directly

without knowing the T-dual expression. Note that the above S-duality transforma-

tion becomes paraticularly simpler when n1 = n2 = · · · = nN :

Z(n,··· ,n)string (ϵ±,m; ρ)

= exp

[−2πi

ρ(m2 − ϵ2+)Nn

]· Z(n1,··· ,nN )

string

(ϵ±ρ,m

ρ;−1

ρ

). (2.125)

The prefactor can be absorbed into a scaling of w′ = q fugacity, conjugate to n.

This expression might be useful to understand the DLCQ of type IIB little strings,

in which U(n)N gauge theory description was used [26].

2.6 Discussion

In this chapter, I explored the 2 dimensional N = (4, 4) and N = (0, 4) gauge

theory descriptions of macroscopic IIA/IIB little strings. In particular, I proposed

a new (0, 4) gauge theory which enables the computation of the IIA strings’ elliptic

genera. We used these elliptic genera to study the little string T-duality.

The elliptic genus is enjoying SL(2,Z) × SL(2,Z) symmetry on the complex

structure τ and Kahler parameter ρ of the torus. Interesting extended dualities

were studied in [43] for 6d maximal SYM theory compactified on T 2, from its

Seiberg-Witten curve. It will be interesting to see whether a larger duality than

what I explored here is realized in the elliptic genera.

It will also be interesting to see if one can study the T-duality of elliptic genera

for the heterotic little string theories, living on the heterotic 5-branes in the SO(32)

and E8×E8 theories. Just like our IIA strings are closely related to the ‘M-strings’

of 6d (2, 0) CFT, the E8×E8 little strings would be closely related to the so-called

E-strings of the 6d (1, 0) CFT, with E8 global symmetry [44, 45]. The E-string

elliptic genera have been recently studied in [46], from 2d (0, 4) gauge theories.

Finaly, the self-dual string elliptic genera in 6d SCFTs turn out to be related

84

to other interesting observables, such as the superconformal indices [47]. It will be

interesting to see if the elliptic genera for little strings also find similar interesting

applications.

85

Chapter 3

3d Seiberg duality and Vortices

The result of this chapter is based on the paper [48].

3.1 Vortices in 3d gauge theories

Vortices are the localized magnetic flux tubes on the 2 dimensional spatial surface,

and they can be regarded as solitonic particles in 3d gauge theories. Quantum

Vortices appears in Higgs phase of gauge theories in which gauge symmetry is

spontaneously broken. For example, we can consider vortices in superconducting

systems. In a superconducting phase below the critical temperature, two electrons

form ‘Cooper pairs’ as the composite bosonic particle, and spontaneously break the

gauge symmetry and generate the mass gap. In the strongly-coupled regime, the

photon is very heavy while the vortices appears as the lightest quantum states.

Generally, 3d gauge theories have the gauge coupling constants with mass

dimension, and the gauge interaction becomes strongly-coupled in IR. There are

many phases in gauge theories with the relevant coupling. For example, we can

consider the confining phase with a dynamical mass-gap or the strongly-coupled

IR conformal phase. Confinement occurs in 4 dimensional QCD system, of which

gauge coupling is marginally-relevant. In IR confining systems, the gauge-singlet

composite particles as mesons and baryons appear due to the strong gauge coupling,

and a mass-gap is dynamically generated. On the other hand, in IR conformal

phase, there is no mass generation, while the arbitrarily long range correlations

mediated by light degrees of freedom occur in the system, in which we can not

apply ‘particle interpretation’ to these excitations. In this chapter, I am mostly

interested in the conformal phase.

It is usually very hard to study the strongly-coupled IR conformal phases

86

directly. One way to study is to investigate the physical aspects near the conformal

fixed point. In this view point, I can study the gauge theories near the IR conformal

fixed point with a small relevant deformation, and dynamics of the light excitations

in this phase. I can expect that the light excitations will become massless degrees

in the very IR fixed point, and mediate the long range correlations.

In the 3d gauge theories with complex scalar fields, there is the Higgs phase in

which the gauge symmetry is spontaneously broken by condensation of the scalar

field. If considering a small gauge-symmetry broken scale, we can regard that the

system is strongly coupled and located near the conformal fixed point. In these

strong-coupling Higgs phases, the vortex solitons occurs as the lightest excitations.

We can expect that these light vortices mediate the long range correlation of the

system in the conformal phase. Therefore, studying the low energy dynamics of

the vortices can be one of the main methods to study physical aspects of these 3d

strongly coupled gauge theories in IR conformal phase.

3.2 Supersymmetric gauge theories and vor-

tices

In this chapter, I study dynamics of vortices in 3d N = 4, 3 supersymmetric gauge

theories.

Supersymmetric gauge theories in 3 dimensions have been extensively studied

with various motivations in recent years. 3d supersymmetric Chern-Simons theories

coupled with matters as superconformal field theories have been studied with mo-

tivation to explore M2-branes’ low energy dynamics in M-theory. In this purpose,

the 3d N = 6 Chern-Simons-matter theories was discovered as M2-branes world-

volume theories. Moreover, various 3d superconformal field theories also have been

studied to test CFT3/AdS4 correspondence. Besides, various strong-coupling dual-

ities among these theories have been found in various contexts. Many theories can

be related to other theories by dualities in strongly coupled phases. This dualities

have been also extensively studied.

87

Progress in understanding 3d supersymmetric field theories is related to a

series of developments in computing physical quantities in strongly-coupled super-

symmetric systems, especially, in 3d superconformal field theories. These physical

quantities preserve a specific subset of supersymmetry in their physical systems,

and they are protected from quantum correction. I can compute these quanti-

ties non-perturbatively by using supersymmetry. This technique of computation is

called as ‘localization method’.

In this context, there have been great progresses regarding 3d SUSY gauge

theories. Especially, the superconformal index [49] was exactly computed for 3d

N = 6 Chern-Simons-matter theories, called as ‘ABJM’ theory [50]. The supercon-

formal index is partition function of S2×S1, which is counting local BPS operators.

With this quantity, CFT3/AdS4 correspondence was seriously tested as duality be-

tween the superconformal field theories on M2-branes and M-theory on AdS4×S7.

Then, the superconformal indices has been also computed various 3d supercon-

formal field theories, and its various applications has been found. The partition

function of 3-sphere have been also exactly computed in various SCFTs. Firstly, it

was computed in N = 6 Chern-Simons-matter theories, and it have been studied to

test CFT3/AdS4 correspondence in superconformal field theories [49]. then broad

applications are studied and generalized in wider class of 3d superconformal field

theories.

I will study another supersymmetric partition function involving interesting

information in dynamics of 3d supersymmetric gauge theories. As I mentioned, 3d

gauge theories in Higgs phase, including the relevant deformation of 3d supercon-

formal field theories, can have non-perturbative solitonic vortex particles in their

spectra. These vortices are important in that these become most lightest degrees

in dynamics near conformal fixed points. In 3d supersymmetric gauge theories,

these BPS vortices preserving some supersymmetry play important roles in clari-

fying existence of the superconformal fixed points [51], and mirror symmetry [19]1,

1Abrikosov-Nielsen-Olesen(ANO) vortices are identified as the confined states of electrons inmirror-dual theory.

88

and etc. I compute and study the supersymmetric partition function counting BPS

bound states of topological vortices. In some theories, there are partially broken

Higgs phase, and admit non-topological vortices, which carrying magnetic charge

and electric charges. Non-topological vortices will be discussed briefly in this thesis.

Using the supersymmetric partition function of the topological vortices, I investi-

gate dynamical aspects in 3d gauge theories, and strong-coupling dualities.

Vortex partition functions was studied in 2d supersymmetric gauge theories.

In this context, vortices play a role of instantons in 2d gauge theories. the partition

function of 3d theories compactified on a circle has been already concerned well.

Since instantons in Euclidean field theories has natural interpretation as solitonic

particles in one more higher dimensions, I can interprete the vortex partition func-

tion of 3d theories as index counting BPS bound states of vortex particles. However,

I will encounter subtleties in index interpretation for the semi-local vortices, which

involve non-compact internal moduli.

Besides, it have been found that some squashed 3-sphere partition functions

can be factorized by vortex partition functions in some theories [52]. Moreover,

the superconformal index which counts the BPS monopole operators is crucially

related to the vortex partition function, since the monopole operators create and

annihilate the nonzero vortex charges. This relation was rigorously tested, and it

was found that superconformal indices in 3d N = 2 gauge theories can be factorized

by vortex partition function and anti-vortex partition function [53].

3d N = 4 gauge theories which I will consider are supersymmetric Yang-Mills

theories coupled with fundamental matters. 3d N = 3 gauge theories have Chern-

Simons interaction, as well as the Yang-Mills interaction.

The vortex partition functions for these theories involve various non-perturbative

dualities. Especially, 3d N = 4 gauge theories have 3d Seiberg dualities. Originally,

Seiberg dualities was found in 4d N = 1 SU(N) gauge theories in the ‘conformal

window’, which is 32N ≤ Nf ≤ 3N , where N is the rank of gauge group and Nf is

the number of flavors. In this regime, 4d N = 1 gauge theories have the IR fixed

point. 4d Seiberg duality means that there exist dual SU(Nf −N) gauge theories

89

coupled with Nf flavors and one mesonic field, which describe the same IR fixed

point. Since all of these 4d gauge theories have the marginal couplings in IR, if

the original gauge theories have the strong gauge-coupling, the dual gauge theories

are in weak-coupling. This duality is very similar to electromagnetic duality. Out

of the conformal fixed points, If the gauge theories in N < Nf < 32N have IR

confining phases, and the mass-gap is dynamically generated. These theories have

Seiberg-dual SU(Nf − N) gauge theories having IR free phases. Seiberg duality

means that the confined gluballs, baryons, and magnetic monopoles in IR confin-

ing phase realized as the dual Higgs mechanism maps to the elementary quarks in

the dual theories2.

However, 3d N = 4 gauge theories have dimensional-full gauge couplings,

and can not have IR coupling coupling constants. These gauge theories are all in

strong-coupling in IR. Even though 3d Seiberg duality has a formal similarity with

4d Seiberg duality, but interpretation of them are quite different. While 4d dualities

can be interpreted as electromagnetic duality, rather, 3d Seiberg duality describe

‘strong-strong’ coupling duality in IR dynamics.

In this chapter, I restrict our study to 3d N =4 and 3 supersymmetric gauge

theories with U(N) gauge group coupled to matter fields in fundamental repre-

sentations. Our partition function can be used to study various non-perturbative

dualities of these theories. I study the 3d Seiberg-like dualities similar to [54].

Seiberg duality is an IR duality, in which two different UV theories flow to the

same IR fixed point. Although it was originally found in 4d N =1 SQCD [16], such

dualities were also discovered and studied in 3 dimensions. 3d Seiberg dualities were

first discussed in [54]. Although they have some formal similarities to 4d Seiberg

dualities, physical implications of these dualities are not quite the same in different

dimensions. 4d Seiberg duality can be regarded as an electromagnetic duality and

also as a weak-strong coupling duality [?]. Similar interpretation in 3d is lacking,

2In Higgs mechanism, electrons condenses, the magnetic fields form the flux tube, and themagnetic interaction are confining. On the other hand, in dual Higgs mechanism, the monopolescondensation occur, and the system behaves electrically confining phase.

90

at least when there are no Chern-Simons term so that I do not have IR couplings.

Seiberg duality also exists after introducing a Fayet-Iliopoulos (FI) deforma-

tion on both sides of the dual pair. Denoting by ζ the FI parameter, there exist

BPS vortex solitons whose masses are proportional to ζ. Considering the regime in

which ζ is much smaller than the Yang-Mills coupling scale g2YM , 3d Seiberg du-

ality should map different types of ‘light’ vortex particles in the dual pair. I show

that the spectra of the topological vortex particles as seen by our partition func-

tion perfectly agree between the Seiberg-dual pairs, when they exhaust all possible

vortices (without non-topological vortices). This is the 3d version of the 4d Seiberg

duality map. While in the latter case glueballs, baryons and magnetic monopoles

in the confining phase map to the elementary quarks in the dual Higgs phase [?],

in 3d I naturally find that vortices map to dual vortices.

3d N =4 Seiberg duality can be partly motivated by brane systems [55]. The

N =4 that I consider in this chapter can be engineered by the D-brane/NS5-brane

system shown in Fig 7. By changing the positions of the two NS5-branes, which

makes them cross each other when ζ = 0, one obtains another 3d gauge theory

with U(Nf −N) gauge group and Nf flavors for Nf ≥ N . Supposing that both

theories flow to nontrivial IR fixed points, Seiberg duality asserts that the two IR

fixed points are the same. However, as pointed out in [51, 18], it turns out that

one of the two UV theories in the putative dual pair often does not flow to an IR

CFT, at least not in the ‘standard way’ [51] in which the SO(4) superconformal R-

symmetry is the SO(4) R-symmetry visible in UV. One way to see this is to study

the R-charges of BPS monopole operators, which I review in section 3. Firstly, when

Nf < 2N − 1, there exist monopole operators whose R-charges are smaller than

12 . If the theory flows to a CFT, the BPS bound demands that this R-charge be

the scale dimension of the operator, violating the unitarity bound. These theories

were called ‘bad’ [51, 18]. When Nf = 2N − 1, called ‘ugly’ case, there exists a

monopole operator with R-charge 12 , saturating the unitarity bound in IR. In this

case, the modified version of the naive Seiberg duality is that the U(N) theory with

Nf = 2N−1 flavors is dual to the U(Nf−N) = U(N−1) theory with Nf = 2N−1

91

flavors times a decoupled free theory of a (twisted) hypermultiplet. This has been

recently tested from the 3-sphere partition function [56]. As the case with Nf = 2N

is trivially self Seiberg-dual, the pair containing the case with Nf = 2N − 1 was

the only nontrivial dual pair [51, 18].

Our vortex partition function confirms this duality at Nf = 2N − 1: namely,

the partition function agrees with that of the U(N−1) theory with Nf = 2N−1 fla-

vors times the vortex partition function of the N = Nf = 1 theory (the Abrikosov-

Nielsen-Olesen vortex). As the last vortices are free, it agrees with the above ar-

gument that the free hypermultiplet sector exists. Also, the monopole operator

mentioned in the previous paragraph with dimension 12 is nothing but the vortex-

creating operator, making it natural to identify the above free hypermultiplet as the

vortex supermultiplet. The test can be made at each vortex number k = 1, 2, 3, · · · ,

which makes the confirmation highly nontrivial. Moreover, our vortex partition

function suggests that the general putative dual pair for any Nf ≥ N could be

actually Seiberg-dual to each other, also with a modification by adding a factor-

ized sector. This may be suggesting a broader class of IR fixed points than those

identified in [51]. See section 4.1 for the details.

Seiberg dualities with N ≤ 3 supersymmetry were also studied quite exten-

sively in recent years, after they were discovered in Chern-Simons-matter theories

[57, 58, 59]. For instance, [18, 60, 61, 62, 63, 64] studied various 3d Seiberg dualities

using the 3-sphere partition function and the superconformal index. Other studies

on the Chern-Simons Seiberg dualities include [65]. I study the vortex partition

function of the above N = 4 theory, deformed by an N = 3 Chern-Simons term.

With nonzero FI deformation, the vacuum structure becomes more complicated

than that for the gauge theory with zero Chern-Simons term, as one also finds

partially Higgsed phases. The Seiberg duality maps a branch of vacuum to another

definite branch in the dual theory. Due to the presence of partially unbroken Chern-

Simons gauge symmetry, it turns out that the study of non-topological vortices is

also crucial for the Seiberg duality invariance of the vortex spectrum. From our

index for topological vortices, I study aspects of the Seiberg-dual non-topological

92

vortices. In some simple cases, our topological vortex index confirms nontrivial

properties of non-topological vortices suggested in the literature via Seiberg dual-

ity. Namely, I show that the vorticity and angular momentum of non-topological

vortices in the Chern-Simons-matter theory with N = Nf = 1 satisfy a bound

required by a tensionless domain wall picture of [66], via topological vortex calcu-

lation of the Seiberg-dual theory.

The remaining part of this chapter is organized as follows. In section 3, I

explain N =4, 3 field theories, BPS topological vortices, and then derive the vortex

partition functions (or indices). In section 3, I show that these partition functions

nontrivially confirm the known Seiberg dualities of some N = 4 theories with FI

deformations. I then suggest a wide extension of this duality, presumably at new

kinds of IR fixed points. I also study N =3 Seiberg dualities from vortices. Section

4 concludes with discussions. Appendices A, B, C explain the structure of vortex

quantum mechanics and also a derivation of the vortex partition function.

3.3 Vortex partition functions of 3d gauge the-

ories

3.3.1 Supersymmetric gauge theories and vortices

Let us first consider the N = 4 U(N) gauge theory with Nf fundamental hy-

permultiplets. Its bosonic global symmetry is SO(2, 1) coming from spacetime,

SO(4) = SU(2)L × SU(2)R R-symmetry which rotates the supercharges as a vec-

tor, and an SU(Nf ) flavor symmetry. It has a vector supermultiplet consisting of

the gauge field Aµ, gaugino λabα (where a, b are SU(2)L and SU(2)R doublet in-

dices, respectively, and α is for SO(2, 1) spinors), and three scalars ϕI (I = 1, 2, 3

for SU(2)L triplet). The hypermultiplets consist of Nf pairs of complex scalars qia

in the fundamental representation of U(N), where i = 1, 2, · · · , Nf , and superpart-

ner fermions ψaiα . The supercharges Qabα are taken to be Majorana spinors involving

93

SU(2)L × SU(2)R conjugations. The bosonic part of the action is given by

Lbos =1

g2YMtr

[−1

4FµνF

µν − 1

2Dµϕ

IDµϕI +1

4[ϕI , ϕJ ]2

]−Dµq†iaDµq

ia − q†iaϕIϕIqia − 1

2g2YMDADA (3.1)

where A = 1, 2, 3 is the triplet index for SU(2)R, and g−2YMD

A = qia(τA) ba q†ib

with three Pauli matrices τA. The moduli space of this theory has two parts. The

classical Coulomb branch is obtained by taking ϕI to be nonzero and all diagonal,

while all hypermultiplet scalars are set to zero. The Higgs branch is obtained by

taking ϕI = 0, while nonzero qia satisfy the condition DA=0. The real dimension of

the Higgs branch moduli space (modded out by the action of gauge transformation)

is 4N(Nf −N). As I shall be mainly interested in the Higgs branch which supports

vortex solitons, and also due to the motivation of studying Seiberg duality, I shall

restrict our studies to the theories satisfying N ≤ Nf . The Coulomb and Higgs

branches meet at least at a point in which all fields are set to zero. They may

meet more nontrivially in the presence of the vacua with partially unbroken gauge

symmetry when Nf ≤ 2N − 1 [51].

One can also introduce Fayet-Iliopoulos deformations for the overall U(1) part

of U(N), which leaves the form of the (bosonic) action as (3.1) but changes the

D-term fields DA to

g−2YMD

A = qia(τA) ba q†ib− ζA , (3.2)

with three real constants ζA. Nonzero FI parameters break SU(2)R to U(1). With-

out losing generality, I can take ζ ≡ ζ3 > 0 and other two to be zero. It will also

be convenient to call qi ≡ qi1 and qi ≡ q†i2. The vacuum condition DA = 0 can be

written as

qiqi = 0 , qiq†i − qi†qi = ζ . (3.3)

The hypermultiplet scalar should be nonzero and totally break U(N) gauge sym-

metry, lifting the Coulomb branch. With ζ > 0, a subspace of the Higgs branch

94

moduli space which will be useful later is obtained by setting qi = 0. The second

equation of (3.3) is then solved by picking a U(N) subgroup of SU(Nf ), and taking

q =(√

ζ 1N×N | 0N×(Nf−N)

), (3.4)

where I view qi as an N×Nf rectangular matrix. The possible embeddings of U(N)

yields a vacuum moduli subspace given by the GrassmannianSU(Nf )

S[U(N)×U(Nf−N)] .

At any point, S[U(N) × U(Nf − N)] =U(N)×U(Nf−N)

U(1) global symmetry remains

unbroken.

With nonzero ζ (>0), there exist BPS vortex solitons on the above subspace

given by qi = 0. The BPS equations can be obtained either from supersymmetry

transformations or by complete-squaring the bosonic Hamiltonian [67], which are

F12 = g2YM (qiq†i−ζ) , (D1−iD2)qi = 0 , k ≡ − 1

2π

∫d2x trF12 (∈ Z) > 0 . (3.5)

I have chosen to study vortices with k > 0, rather than anti-vortices. The BPS

mass of the vortices is given by 2πζk. There is a moduli space of the solution, with

real dimension 2kNf [67]. These vortices preserve 4 real supersymmetries Qa1− and

Qa2+ ∼ ϵab(Qb1− )† among the full N = 4 supercharges Qabα , where ± denote SO(2, 1)

spinor components in the eigenspinor basis of γ0. The vortex quantum mechanics

model which I introduce later explicitly preserves Qa1− , which shall be written as

Qa.

The nature of topological vortices depends on whether Nf = N or Nf > N .

When Nf =N , the 2kN dimensional vortex moduli space consists of 2k translation

zero modes of k vortices and 2k(N−1) internal zero modes. The N−1 complex zero

mode per vortex can be understood as the embedding moduli of U(1) Abrikosov-

Nielsen-Olesen (ANO) vortex into U(N). Namely, the internal moduli of a single

vortex is CPN−1 = U(N)U(N−1)×U(1) . When Nf > N , 2k(Nf−N) extra internal zero

modes exist. There are non-compact directions from these extra modes, as vortices

can now come with size moduli. These vortices are called semi-local vortices.

95

The low energy dynamics of these vortices can be studied in various ways. It

can be studied by a D-brane realization of the QFT and vortices [67], as I shall

review shortly. Also, one can perform a careful moduli space approximation in

the field theory context, which has been done in [68, 69]. It turns out that some

of the dynamical degrees kept in the naive D-brane considerations [67] originates

from non-normalizable zero modes [68, 69] from the field theory viewpoint. More

concretely, supposing that I introduce an IR cut-off regularization of length scale

L, it was shown that the mechanical kinetic terms for the last modes pick up a

factor proportional to logL [68, 69]. After carefully redefining variables in a way

that IR divergence does not appear, it was shown in the single vortex sector that

the Kahler potential for the quantum mechanical sigma model differs from that

derived from the D-brane approach [68].

Let us explain the difference in some detail and clarify our viewpoint on the

index calculation. Our claim is that the index will be the same no matter which

vortex quantum mechanics is used, as the index is insensitive to various continuous

parameters of the theory. If one could find a continuous supersymmetric deforma-

tion between the string-inspired model of [67] and more rigorously derived field

theory models, this would prove our claim. Actually at k = 1, the two Kahler

potentials of [67] and [68] can be written as

KHT =√r2 + 4r|ζ|2 − r log

(r +

√r2 + 4r|ζ|2

)+ r log(1 + |zi|2) , (3.6)

KSVY = |ζ|2 + r log(1 + |zi|2) (3.7)

with |ζ|2 ≡ (1+|zi|2)|zp|2, where the summations over i range in 1, 2, · · · , N−1, and

those over p range in N, · · ·Nf−1. Deforming the former Kahler potential to the

latter one in a continuous way will prove that there is a supersymmetric deformation

between the two. Of course there is an issue on the non-compact region. As I shall

illustrate with detailed calculations in the appendices, our index can be completely

determined from the information of the moduli space near the region where zp = 0,

96

NS5NS5'

Nf D5's

Nf-N D3's

N D3's

NS5

Nf D5's

Nf-N D3's

N D3's

x6

x3, x4, x5

x7, x8, x9

NS5'

D1 D1

Figure 7: The brane construction of N = 4 Seiberg-dual pairs with nonzero FIparameter ζ. The red lines denote D1-brane vortices.

at which the vortex sizes are minimal. So I can ignore any possible difference in

the asymptotic behaviors of the two metrics. By inserting ϵ|ζ|2 to all |ζ|2 in KHT,

and also multiplying ϵ−1 to the first two terms of KHT, one obtains a 1-parameter

deformation of the Kahler potential. By taking the ϵ→ 0 limit, one finds that KHT

reduces to the exact field theory result KSVY.

Although the above kind of comparison can be made only when the moduli

space metric is explicitly known, I expect the same phenomenon to appear for

multi-vortices. This is because our index is only sensitive to the region near min-

imal size semi-local vortices, and all concrete studies from QFT [68, 69] suggest

that the difference between the two approaches will be absent in this region. In

particular, the second reference in [69] discusses this point for some multi-vortex

configurations. I also mention that [68] finds certain BPS spectrum of the two mod-

els agree with each other. In the rest of this chapter, I shall be working with the

models like [67] derived from the naive D-brane pictures, to derive the index.

One can engineer the above gauge theories and vortices from branes in type

IIB string theory, as shown in Fig 7. The D3-branes, NS5-branes, D5-branes are

along 0126, 012345, 012789 directions, respectively. When ζ = 0, N D3-brane

segments connect two NS5-branes and can move in the 345 direction along the

NS5-brane worldvolume. This forms the Coulomb branch, with R3 showing the

97

SU(2)L symmetry. Extra Nf D3-branes connect the NS5-brane on the right side

(NS5′) and the Nf D5-branes. The open strings connecting the two sets of D3-

branes provide the fundamental hypermultiplet matters. The Seiberg duality that

I shall explore in this chapter corresponds to moving the two NS5-branes across

each other. By the brane creation effect [55], the number of D3-branes between the

two NS5-branes after this crossing is Nf−N .

Turning on nonzero FI term corresponds to moving one NS5-brane along its

transverse 789 directions, parametrized by the SU(2)R triplet ζA. The N D3-branes

cannot finish on NS5′-brane preserving supersymmetry. So one has to combine

them with N of the Nf flavor D3-branes, as shown on the left side of Fig 7. The

remaining Nf −N D3-branes connect the NS5′-brane and the D5-branes. Fig 7

shows the two brane configurations after I move two NS5-branes across them in

the 6 direction. In both cases, there exist BPS D1-branes (as shown by the red

segment), corresponding to the BPS vortices.

It will also be helpful to understand the supersymmetry preserved by these

branes. The NS5-, D5-, and D3-branes preserve supersymmetries which satisfy

definite projection conditions for (σ3)⊗Γ012345, (σ1)⊗Γ012789 and (iσ2)⊗Γ0126 [70,

71], where Γ0123456789 = +1 from type IIB chirality. All three projectors commute.

From

(σ3)⊗ Γ012345 = −[(σ1)⊗ Γ012789

]·[(iσ2)⊗ Γ0126

], (3.8)

two projection conditions imply the third. So one finds a 1/4-BPS configuration,

preserving 8 real or 3d N =4 SUSY. More concretely, the 8 SUSY may be obtained

as follows. Taking the 6 commuting matrices, A = (iσ2) ⊗ Γ0, B = Γ12; C = Γ34,

D = (σ1)⊗ Γ5; E = Γ78, F = (σ3)⊗ Γ9, one can write Γ6 = −ABCDEF and also

write the 3 projectors as

ΓNS5 = +ABCD , ΓD5 = −ABEF , ΓD3 = ABΓ6 = +CDEF , (3.9)

respectively. The 32 real components of the type IIB spinor can be obtained

98

by starting from 64 dimensional real spinor (with 32 components from 10d and

2 components from SL(2,R) Pauli matrices), and subjecting them to the chi-

rality condition. However, the matrices A,D,F do not commute with the chi-

rality operator Γ11. So to obtain the eigenspinors of the BPS projections using

A,B,C,D,E, F eigenstates, one would always have to make a linear combination

of different eigenstates of A,D,F at the final stage, to make them eigenstates of

Γ11. Supposing that 1st/3rd projectors for NS5/D3-branes come with +1 eigen-

values, one has AB = ±i, CD = ∓i, EF = ±i, where the ± signs are corre-

lated. The possible signs of the eigenvalues and eigenvectors of (A,B,C,D,E, F )

come in 8 cases Ψs1,s2,s3 ∼ (s1, s1; s2,−s2; s3, s3) for the upper signs, and 8 cases

Υs1,s2,s3 ∼ (s1,−s1; s2, s2; s3,−s3) for the lower signs, where s1, s2, s3 are indepen-

dent ± signs. Since the A,D,F do not commute with Γ11 but rather anticom-

mute, I should mutiply 1+Γ11

2 to the spinors to get 10d chiral spinors. I can de-

fine Γ11Ψs1,s2,s3 ≡ Υ−s1,s2,s3 . The chirality projection only keeps the combination

Ψs1,s2,s3 +Υ−s1,s2,s3 , and I finally have 8 or 3d N = 4 SUSY.

The supersymmetry for D1-brane vortex is given by the projector (σ1)⊗Γ09 =

−AF [71], supposing that the FI parameter is separating two NS5-branes along the

9 direction. This again commutes with the remaining two projections, and makes

the vortex preserve 4 real SUSY. More concretely, let us assume that AF has +1

eigenvalue. In the above two classes of ± sign, I can take either A = ±1, F = ±1

before Γ11 projection, yielding (s1, s1; s2,−s2; s1, s1) or (s1, s1; s2, s2;−s1, s1). Thus,

one has 8 spinors before projection, and after Γ11 projection one obtains 4 SUSY.

As the two NS5-branes cross, the D1-brane in the U(N) theory and U(Nf − N)

theory appears as vortex/anti-vortex, respectively, depending on whether the D-

string starts or ends on the brane on which the 3d gauge theory lives. I should thus

compare the vortex and anti-vortex spectra of the two theories.

One can also study vortices in the N = 3 theory with an FI deformation. I first

review the N = 3 theory with FI term and its vacua. One obtains the N = 3 Yang-

Mills Chern-Simons gauge theory by adding a Chern-Simons term to the above

N = 4 theory. Keeping the three D-term fields DA off-shell, one adds to the action

99

the following Chern-Simons term

κ

4π

∫tr

(ϵµνρ(Aµ∂νAρ −

2i

3AµAνAρ)− 2ϕADA

)+ fermions . (3.10)

The SU(2)L×SU(2)R R-symmetry of the N = 4 theory is reduced to the diagonal

SU(2). So I no longer distinguish the I and A triplet indices of two SU(2)’s, or the

dotted/undotted doublet indices. By integrating out DA, one obtains the bosonic

potential

− 1

4g2YM[ϕA, ϕB]2 +

1

2g2YMDADA , g−2

YMDA = qia(τA) ba q

†ib− ζA − κ

2πϕA . (3.11)

The classical supersymmetric vacuum solutions can be found from the above

bosonic potential. One first finds that the Coulomb branch is lifted. This is because

ϕA acquires nonzero mass either from a superpartner of the Chern-Simons term, or

by the Higgs mechanism. With ζ = 0, one finds many partially Higgsed branches.

For simplicity, I shall only consider the subspace in which qi = 0 which admits

topological BPS vortices. The classical supersymmetric vacuum with qi = 0 can be

obtained from the following equations,

qq† − ζ − κ

2πσ = 0 , σq = 0 , (3.12)

where I have set ϕ1 = ϕ2 = 0 (with σ ≡ ϕ3) as they will also be zero when the FI

term is along the third component ζ ≡ ζ3 only. The simplest solution is obtained

by setting σ = 0. Then the condition qq† = ζ simply yields theU(Nf )

U(N)×U(Nf−N)

moduli space. More generally [?], I can take an n × n block of the N ×N matrix

σ to be nonzero. Then from the second condition, q has to sit in the (N − n)×Nf

block orthogonal to σ. From the first equation, qq† and − κ2πσ are rank N − n and

n matrices, respectively, which are acting on mutually orthogonal subspaces. Thus

100

NS5 (1, )

Nf D5's

Nf-N+n D3's

n D3's

N-n D3's

NS5(1, )

Nf D5's

Nf-N+n D3's

- n D3's

N-n D3's

x6

x3, x4, x5

x7, x8, x9

D1 D1

F1 F1

Figure 8: The brane construction of N = 3 Seiberg-dual pairs and vortices withunbroken U(n)κ or U(k − n)−κ gauge symmetry.

the solution of the first equation is

σ = −2πζ

κ1n×n , qq† = ζ1(N−n)×(N−n) → q ∈

U(Nf )

U(N − n)× U(Nf −N + n).

(3.13)

The U(N)κ gauge symmetry is broken by the above vacuum to U(n)κ. From

the quantum dynamics of this U(n)κ Chern-Simons gauge theory, supersymmet-

ric vacua exist only when 0 ≤ n ≤ κ [72]. Therefore, there are min(κ,N) + 1

branches of partially Higgsed supersymmetric vacua, labeled by n in the range

0 ≤ n ≤ min(κ,N).

It is helpful to consider all these aspects from the brane construction, as shown

in Fig 8. The situation is similar to the N = 4 brane configuration, but to induce

nonzero Chern-simons term, one changes the second NS5-brane to an (1, κ) 5-

brane tilted in the 345 and 789 direction. Namely, apart from the 012 direction,

the worldvolume of the (1, κ)-brane has to be aligned along x3 + tan θx7, x4 +

tan θx8, x5+tan θx9 directions with tan θ = κgs, where gs is the type IIB coupling

(at zero RR 0-form, which I assume for simplicity). Putting nonzero FI term ζA

again corresponds to moving the (1, κ) brane relative to the NS5-brane in the 789

direction. When ζ = 0, again there are N D3-branes connectiong NS5- and (1, κ)-

branes, and also Nf D3-branes connecting (1, κ)- and D5-branes. When ζ = 0, there

101

are many possible deformations of this D3-brane configurations, corresponding to

various partially Higgsed phases. On the left side of Fig 8, there can be some

fraction of N D3-branes which can connect NS5- and (1, κ)-branes even after FI

deformation. I take n D3-branes to do so. The remaining N−n of them should

combine with the flavor D3-branes as shown in the figure, whose gauge symmetry

is spontaneously broken. The brane configuration maps to the partially Higgsed

branch with unbroken U(n)κ gauge symmetry. The proposed Seiberg duality [57]

is obtained by moving NS5- and (1, κ)-brane across each other. One then obtains

a U(Nf −N + |κ|) theory coupled to Nf fundamental hypermultiplets, at Chern-

Simons level −κ. With brane creations [55], the vacuum on the left side of Fig 8

maps to the branch in the dual theory with unbroken U(κ−n)−κ gauge symmetry,

on the right side of the figure.

The BPS vortices in the N = 3 theory appear in many different ways. I first

consider them in the brane picture. Firstly, there can be D1-branes connecting the

N −n D3-branes, corresponding to the broken U(N −n) gauge symmetry, and the

Nf −N + n D3-branes corresponding to the remaining flavor branes. See the red

horizontal line on the left side of Fig 8. As the D1-brane ends on the N−n D3-brane

for broken gauge symmetry, they would correspond to topological vortices, similar

to the vortices in the N = 4 theories. Actually, there exist 4d states given by this

segment of D1-brane freely moving along the D3-branes, behaving as monopoles in

the decoupled 4d gauge theory. So the D1-branes would be visible in the 3d theory

as vortices only when they are marginally bound to the 5-brane, as shown in the

figure.

There can also be vertical massive fundamental strings connecting the n D3-

branes (with unbroken U(n)κ symmetry) and other D3-branes. If the FI parameter

deformation is made in the 9 direction, the string is stretched in the 5 direction.

I shall shortly show that this configuration preserves same SUSY as the D1-brane

vortices. Also, as U(n)κ gauge symmetry is unbroken, the overall U(1) Noether

charge (electric charge) induces nonzero∫trU(N)F12 vorticity via the Gauss’ law

with U(n)κ Chern-Simons term. From the field theory perspective, these vortices

102

are often called non-topological vortices.

Also, there can be strings made of one D1 and κ F1’s, which vertically end on

the (1, κ) 5-brane and N−n D3-branes. This configuration preserves the same SUSY

as the above two types of vortices. One can also show that the energy of this string

is exactly the same as the D1-brane vortex of first type, i.e. 2πζ, by calculating the

length and tension of the string. It seems that our topological vortex index should

be counting these configurations as well.

Although I follow [67] to consider theN =3 version of their brane configuration

given by Fig 8, it is often clearer to move Nf D5’s along x6 to have it between the

other two 5-branes [57]. See section 3.2 for more explanations.

It is easy to check the supersymmetry of these brane configurations. The NS5-,

D5-, (1, κ)- and the D3-branes require the projection conditions for (σ3)⊗ Γ012345,

(σ1)⊗Γ012789, (cθσ3+ sθσ1)⊗Γ012(cθ3+sθ7)(cθ4+sθ8)(cθ5+sθ9), (iσ2)⊗Γ0126. To study

the common eigenstates of these projectors, I again express all the projections

in terms of the six commuting projectors A,B,C,D,E, F . The eigenstates of 3

projectors which are inherited from the N = 4 theory can again be solved in

terms of 8 spinors Ψs1,s2,s3 = (s1, s1; s2,−s2; s3, s3) and 8 other spinors Υs1,s2,s3 =

(s1,−s1; s2, s2; s3,−s3) before chirality projection. The projection for the (1, κ)-

brane is given by

AB(c2θD − s2θF − isθcθσ2(D + F )

) (c2θC + s2θE + sθcθΓ

38(1 + CE)). (3.14)

One way for this projector to have +1 eigenvalue is to have DF = −1 (real2), CE =

−1 (imaginary2) so that both parentheses in the projector yield ±1, independent

of θ. In this case, one obtains from the above 16 spinors the following 8 cases:

Ψs1,s2,s2 = (s1, s1; s2,−s2; s2, s2) or Υs1,s2,s2 = (s1,−s1; s2, s2; s2,−s2). One also

has to demand that the projection for the (1, κ)-brane comes with a definite sign.

is2 is the last factor including C,E, and ∓s2 is the second factor including D,F ,

where ∓ is for the Ψ/Υ cases. So the 2nd times 3rd factor becomes ∓i. Since AB

in the 1st/2nd case is ±i, this cancels with the ∓i to always yield +1. So I have 8

103

components of spinors before chirality projection. The chirality projection demands

the combination Ψs1,s2,s2 +Υ−s1,s2,s3 , leaving 4 SUSY.

There is a different way of having (3.14) satisfied. Rather than having the

second/third parenthesis to be separately θ independent numbers, the two factors

can yield θ dependent expression which cancel each other. So I start by assigning

D = F = s, which would yield c2θD − s2θF − isθcθσ2(D + F ) = se−2iθσ2 . Assigning

definite eigenvalues for D,F is possible as the third factor does not change their

eigenvalues. σ2 operator changes the eigenvalues of D,F but leaves all other eigen-

values unchanged. Now take C = −E = is′. Then the last factor becomes is′e2θΓ38.

The matrix Γ38 changes the C,E eigenvalues while leaving all the other eigenval-

ues unchanged. The matrix e−2iθσ2e2θΓ38

can be diagonalized by suitably mixing

two states in Ψ,Υ with different signs s2, s3. Since I am restricted to the sector

D = F , C = −E, I only consider Ψs1,s2,−s2 and Υs1,s2,−s2 . The matrix e−2iθσ2e2θΓ38

is expanded as

cos2 2θ + sin2 2θ(−iσ2Γ38) + sin 2θ cos 2θ(Γ38 − iσ2

). (3.15)

The last linear terms are taking states out of the subspace which satisfies the

NS5-, D5-, D3-brane projections. So these terms should vanish by canceling with

each other. This freezes the linear combination of Ψs,+,− and Ψs,−,+, and also

that of Υs,+,− and Υs,−,+. The remaining −iσ2Γ38 is also diagonalized then, with

eigenvalue +1, making the whole projection to be +1. I thus have two Ψ type states

with two values for s1, and similarly two Υ type states. The chirality projection

again relates Ψ and Υ type spinors, so that I am left with 2 SUSY from this sector

labeled by s1. Collecting all, one obtains 6 or 3d N = 3 SUSY.

Considering the D1-brane projection, again I take A = ±1 and F = ±1 com-

ponents. From the first 4 SUSY of the N = 3 theory, one obtains (s, s; , s,−s; s, s)

or (s,−s;−s,−s;−s, s) with s = ±, obtaining Ψs,s,s + Υ−s,s,s after the chirality

projection. However, in the last set of 2 SUSY of the N = 3 theory, note that dif-

ferent F eigenstates are all mixed up for given value of A eigenvalue. As this makes

104

it impossible to correlate the signs of A and F eigenvalues, D1-branes cannot pre-

serve this part of SUSY. So I have D1-brane vortices preserving 2 SUSY. SUSY of

fundamental string vortices can be studied similarly. Its projection σ3 ⊗Γ05 = AD

demands A = −D = ±1, where the relative minus sign is chosen to stay in the

same BPS sector as D1-branes. From 4 of N = 3 SUSY, one obtains Ψs,s,s+Υ−s,s,s,

which are the same 2 SUSY as those for D1-branes. From 2 of the N = 3 SUSY,

again no further SUSY appears.

From field theory, the supersymmetry of the N = 3 theory is obtained by

restricting the N = 4 SUSY by identifying SU(2)L, SU(2)R, and taking the same

off-shell (for three D-term fields) SUSY for Qabα for symmetric a, b. Equivalently,

one can write the supercharges as QAα . The 2 SUSY preserved by our vortices take

the form of Q11− ∼ (Q22

+ )†, and this will be the same 2 supercharges that I will

use to calculate the index even in the N = 4 theories. The BPS equations for the

N = 3 topological vortices are the same as those for the N = 4 vortices.

The fundamental strings discussed above should be distinguished from the

topological vortices in classical field theory, as the so-called non-topological vor-

tices. However, only the total spectrum of all vortices will have a duality invariant

meaning in partially unbroken phases. Non-topological vortices are discussed in

the literatures: for instance, see [73] and references therein. In particular, non-

topological vortices in supersymmetric Maxwell-Chern-Simons theories are studied

in [74]. The mass is given by the electric charge in the unbroken phase multiplied

by the mass of an elementary particle [74], supporting that they are bounds of

fundamental strings.

3.3.2 Vortex quantum mechanics

I review the quantum mechanical description of topological BPS vortices in the

N = 4 theory [67] motivated by branes, and also explain how to include the ef-

fect of nonzero Chern-Simons term preserving N = 3 supersymmetry [?, 75]. As

explained before, it has been discussed [68, 69] that some of the degrees in this me-

105

chanics come from non-normalizable zero modes of the soliton, demanding special

care about IR regularization to correctly understand their low energy dynamics

[68]. As concretely supported with single vortices and generally argued in the pre-

vious subsection, I think the difference between the two mechanical models will not

affect the index that I calculate and study, by having two models connected by a

continuous supersymmetric deformation (zooming into region of the moduli space

with minimal sizes).

In the N =4 theory, the 4 supercharges Qa1− (and the conjugate Qa2+ ) preserved

by the vortices appear as the supercharges of the mechanical model. I call Qa ≡ Qa1−

in the mechanics. The SU(2)L global symmetry (with a doublet index) is manifest.

As explained in [67], the dynamical degrees of this mechanics can be obtained by

a dimensional reduction of 4d N = 1 superfields down to 1d, regarding the above

SU(2)L as the internal 3d rotation in the 4d to 1d reduction. SU(2)R in the 3d QFT

is broken by the FI term to U(1)R. As the hypermultiplet scalar qi ≡ qi1 assumes

nonzero expectation value, the surviving U(1) is a linear combination of U(1)R and

the overall U(1) of U(N) gauge symmetry which leave the VEV invariant. I simply

call the last combination U(1)R.

The gauged quantum mechanics for k vortices has the following degrees: N

chiral multiplets qi, ψia in the fundamental representation of U(k), Nf −N chiral

multiplets qp, ψap in the anti-fundamental representation of U(k), a chiral multiplet

Z, χa in the adjoint representation of U(k), and the U(k) vector multiplet At, ϕI , λa.

The variables qi and qp should not be confused with complex scalar fields in 3d

QFT. In fact, the moduli coming from these mechanical variables all originate from

106

the zero modes of q fields in QFT. The Lagrangian is given by [67]

L =tr

[1

2DtϕIDtϕI + |DtZ|2 + |Dtq|2 + |Dtq|2 + iλaDtλa + iχaDtχa

+ iψaDtψa + i¯ψaDtψa +

1

4[ϕI , ϕJ ]

2 − |[ϕI , Z]|2 − qq†ϕIϕI − q†qϕIϕI

− 1

2

([Z,Z†] + qq† − q†q − r

)2+ λa(σI)ab[ϕI , λ

b] + χa(σI)ab[ϕI , χb]

+ ψa(σI)abϕIψb − ¯

ψa(σI)abψbϕI +

√2iχa[λa, Z] +

√2i[Z†, λa]χa

+√2i(qψaλa + λaψaq

† − q†ψaλa − λa¯ψaq

)], (3.16)

where all SU(2)L doublet indices are raised/lowered by ϵab, ϵab. The N chiral mul-

tiplet fields are regarded as k ×N matrices, while Nf −N of them with tilde are

regarded as (Nf −N)× k matrices. r is proportional to the inverse of 3d coupling

constant, 1g2Y M

. The supersymmetry and other properties of this model is summa-

rized in Appendix A. The classical solution for the ground state is given by taking

the D-term potential to vanish,

[Z,Z†] + qiq†i − qp†qp = r . (3.17)

In the D-brane realization, the sign of r depends on the relative position of the

two NS5-branes in Fig 7. On the left side of the figure, the vortex mechanics for

the corresponding 3d theory has r > 0. On the right side, r < 0 for the putative

Seiberg-dual theory. The moduli spaces of the vortices are different for r ≷ 0, but

their real dimensions are all 2Nfk.

The effect of nonzero Chern-Simons term to this mechanics is investigated in

[76], and more recently in [?, 75]. To the above gauged quantum mechanics, I add

the following term [75]

∆L = κ tr(At + ϕ) (3.18)

where ϕ ≡ ϕ3 is the component of the vector multiplet scalar along the nonzero FI

parameter ζ = ζ3. (3.18) is argued to encode the correction in the moduli space

107

dynamics to the leading order in κ [?, 75]. So this model should be reliable (of

course modulo the non-normalizable mode effects) when the Yang-Mills mass scale

κg2YM is much smaller than the FI mass scale ζ. Again, the Witten index I study

in this chapter does not depend on such continuous parameters, which justifies our

usage of this model for calculating the index.

The term (3.18) breaks 4 SUSY of the N = 4 vortices to 2, as it should for our

N = 3 vortices. To see this, recall the supersymmetry transformation of appendix

A,

QaAt = iλa , QaAt = −iλa , QaϕI = i(τ I) ba λb , Qaϕ

I = i(τ I) ba λb .

The term (3.18) only preserves Q2 ∼ Q1 and complex conjugate Q1, since (τ3) 1

1 =

−(τ3) 22 = 1. (3.18) also breaks SU(2)L to U(1), which should happen as the two

SU(2) R-symmetries are locked in the N =3 theories, broken to U(1)R by the FI

term.

Perhaps it is also worthwhile to emphasize that this model was originally

considered in [?, 75] as vortex quantum mechanics of N = 2 theories. At the level

of classical field theory, the difference of the N = 2 theory considered there and

our N = 3 theory is that the latter has an extra term coming from a nonzero

superpotential which couples ϕ1 + iϕ2 to qq. Any possible difference in the vortex

moduli space dynamics coming from this superpotential should appear always with

the 3d field qi, which are always set to zero for classical vortex solutions. Thus, the

bosonic part of the quantum mechanics (consisiting of the vortex zero modes) will

never be affected. The only possible issue is the fermionic term proportional to κ,

which may be added in the case of N = 3 vortices, separately preserving the same

2 SUSY. This will be a well-defined problem which can be studied with the SUSY

transformation of appendix A.

Although I have not carefully studied this possibility, the overall coefficient

of the extra fermionic term is not constrained by the 2 SUSY of vortex quantum

mechanics only. So I should be able to deform the mechanics model in a continuous

108

q ψa q ψa Z χa QaSO(2, 1) 0 −1/2 0 −1/2 1 1/2 −1/2

U(1)R R R+ 1/2 R R+ 1/2 0 1/2 1/2SU(2)L 0 ±1/2 0 ±1/2 0 ±1/2 ±1/2

U(N) N N 1 1 1 1 1U(Nf −N) 1 1 N N 1 1 1

Table 7: Global charges of mechanical variables for N = 4 vortices

way preserving supersymmetry, turning off this term. Then, the possible difference

will not affect the index I study. However, there could possibly be an important

difference between the two models, as the two quantum mechanics models forN = 2

and 3 theories may come with different values of U(1)R charges. This ambiguity

appears because the mechanics only has a D-term potential without an F-term

potential. The value of this charge R for the U(k) fundamental variables qi, qp are

left undetermined in the index calculation. For N = 3 vortices, I should plug in the

canonical value R = 12 inherited by the zero modes of 3d fields. For N = 2 vortices,

there is a possible anomalous shift of R in 3d matter fields, which is meaningful at

least at the conformal point with ζ=0. If one studies N =2 vortices to probe the

physics at the conformal point, it may be important to take the R-charge as that

of the IR CFT with ζ = 0. In this chapter, I only consider the N = 3 version of

the index.

3.3.3 N = 4 and 3 indices for vortices

To define and study a Witten index partition function for topological vortices, I

discuss the symmetries of the vortex quantum mechanics in more detail. Consider

the N = 4 vortex first. The SU(2)L of the mechanics is inherited from the 3d QFT.

I denote its Cartan by JL, whose values for mechanical variables are given in Table

7. Our convention is that the upper a = 1 component has JL = +12 , and so on.

There is also an SO(2) symmetry which rotates Z with charge 1. As the diagonals

of Z roughly correspond to k positions of vortices, I consider it as the rotational

109

symmetry of the 3d theory in SO(2, 1). I call this charge JE , whose values are

listed on the first row of Table 7. The charges for q, q are taken to be zero because

they come from the internal zero modes. Once the charges of bosonic variables are

determined, their superpartners’ charges are fixed by noting that Qa comes from

Qa1− of 3d QFT, which has JE = −1/2. Finally, I consider U(1)R charge JR which is

inherited from the unbroken Cartan of SU(2)R. I want Z to be neutral. q, q† subject

to the mechanical D-term constraint form the internal moduli space of vortices. In

3d solitons, they appear partly from the U(1) embedding of the ANO vortex into

U(N) (for q’s), and also because asymptotic VEV for hypermultiplet fields can be

different from the value at the core of each vortex (for q†). So in QFT, these moduli

all come from the N × Nf fundamental hypermultiplets (which I also called q in

3d), by decomposing them into N × N and N × (Nf − N). From the unbroken

global symmetry, it seems clear that k ×N scalar qi and k × (Nf −N) scalar qp†

in mechanics should have same JR charge. So in Table 7, I naturally set R = −R.

I shall mostly keep R, R as unfixed parameters in general considerations, but at

various final stages set R = −R.3 Furthermore, from the fact that this U(1)R is

inherited from 3d U(1)R ⊂ SU(2)R, I expect R = 12 for N = 4, 3 theories.

Now I consider the Witten index

Ik(µi, γ, γ′) = Trk

[(−1)F e−βQ

2e−µ

iΠie−2iγJe−2iγ′J ′]

(3.19)

for N = 4 vortices, where J ≡ JR+JL+2JE , J′ = JR−JL. This index counts states

preserving Q1 in mechanics, or Q11− in QFT. JR + JL appearing in J is an N = 2

R-charge, which is the first 12 plane rotation in SO(4). JR−JL chemical potential

γ′ has to be turned off when I try to understand the N = 2 SUSY structure of

the index, and also for N =3 vortices. The trace is taken over the Hilbert space of

all single- or multi-particle states with vorticity k. β is the usual regulator in the

Witten index and does not appear in Ik. Finally, Πi for i = 1, 2, · · · , Nf are the

3It happens that the remaining value R will never appear in theN =4 index, assuming R = −R.For N =3 vortices, the index will depend on R even after setting R = −R.

110

S[U(N) × U(Nf − N)] Cartan charges, subject to the condition that∑

iΠi is a

gauge symmetry. In the mechanical model, this overall U(1) is absorbed into the

overall U(1) of U(k) gauge symmetry.

Considering the Euclidean path integral expression for the above index, the

chemical potentials γ, µi provide regulating mass terms for the zero modes of the

vortex mechanics. γ is well known as the Omega deformation of the spatial rotation.

The index interpretation of the γ dependent part is well understood. In particular,

the degree of divergence of each term of the index as one takes γ → 0 is naturally

interpreted as the particle number of the states. See a detailed explanation of [32]

in the context of 5d instanton bound state counting, which applies to our case as

well. The µi dependent part of the index however seems subtle and needs a proper

interpretation, as they correspond to internal zero modes. I do not have a good

physical interpretation at the moment. See the later part of this subsection for a

more detailed explanation on why it is subtle for Nf > N .

This index can be calculated by using localization technique [77], similar to

that used to calculate the instanton partition functions in 4d or 5d gauge theories.

In appendix, I explain a slightly unconventional calculation, which is perhaps a bit

more straightforward in that there is no need for a contour prescription appearing

in ‘standard’ calculations. Of course I shall also view our result in the standard

context, using contour integrals. The localization calculation consists of identifying

the saddle points, and then calculating the determinants around them. In appendix,

I illustrate the calculation for k = 1. I also checked the formulae below for some

higher k’s.

The saddle points for the k vortex index in our calculation are labeled by

the so-called one dimensional N -colored Young diagrams with box number k. It is

obtained by dividing k into N different non-negative integers

k = k1 + k2 + · · ·+ kN , (3.20)

where N non-negative integers ki are ordered. The index contribution from the

111

saddle point (k1, k2, · · · , kN ) is given by

I(k1,k2,··· ,kN )

=

N∏i=1

ki∏s=1

N∏j=1

sinhEij−2i(γ−γ′)

2

sinhEij

2

Nf∏p=N+1

sinhE′

ip−2i(γ+γ′)(R+R)+2i(γ−γ′)2

sinhE′

ip−2i(γ+γ′)(R+R)

2

(3.21)

where

Eij = µi − µj + 4iγ(kj − s+ 1) , E′ij = µi − µj − 4iγ(s− 1) . (3.22)

This expression also admits a contour integral expression:

Ik =1

(2i)kk!

∮ k∏I=1

[dϕI2π

N∏i=1

sinh ϕI−µi+2i(γ+γ′)R−2i(γ−γ′)2

sinh ϕI−µi+2i(γ+γ′)R2

×Nf∏

p=N+1

sinhϕI−µp−2i(γ+γ′)R+2i(γ−γ′)

2

sinhϕI−µp−2i(γ+γ′)R

2

∏I =J

sinhϕIJ2

×∏I,J

sinh ϕIJ+2i(γ+γ′)2

sinh ϕIJ+4iγ2 sinh ϕIJ−2i(γ−γ′)

2

. (3.23)

The integration contour has to be carefully chosen so that only a subset of residues

in the integrand are kept. Introducing zI = eϕI , the contour for zI takes the form of

a closed circle. The simplest possible choice might have been a unit circle surround-

ing the origin zI = 0, regarding ϕI as i times a 2π periodic angle. The contour is

actually more complicated than this. It has to be chosen in a way that the poles

coming from the∏Nf

p=N+1 product of (3.23) all stay outside the contour circle. Also,

the poles from sinh ϕIJ−2i(γ−γ′)2 on the second line as well as poles at zI = 0 coming

from dϕI =dzIzI

are taken outside the contour. Such an exclusion of some residues is

also familiar in the instanton calculus with complicated matter contents. I explicitly

checked this statement on the contour for some low values of k and N,Nf .

By carefully considering the above contour integration expression, one can de-

compose this index to various contributions from different N =2 supermultiplets.

112

Firstly, N =4 vector multiplet combines with the N×N part of the N×Nf hyper-

multiplets (which assume nonzero asymptotic VEV) to yield a basic contribution.

In the N = 2 language, these contributions can be decomposed into those from one

vector supermultiplet, one adjoint chiral multiplet (participating in the N = 4 vec-

tor multiplet), and N2 extra fundamental chiral multiplets and anti-fundamental

chiral multiplets. The contributions are given by

zv =

N∏j=1

sinhE′

ij

2

sinhEij

2

, zadj =

N∏j=1

sinhEij−2i(γ−γ′)

2

sinhE′

ij−2i(γ−γ′)2

, (3.24)

zNfund =

N∏j=1

1

sinhE′

ij

2

, zNanti =

N∏j=1

sinhE′ij − 2i(γ − γ′)

2. (3.25)

The index contribution from this sector is the product of all these four factors. The

‘antichiral’ part denotes contribution from the N×N block of the anti-fundamental

superfields qi, which contribute only to the fermion zero modes without bosonic zero

modes.4 Consider the combinations of the two contributions zvzNfund and zadjz

Nanti:

zvzNfund =

N∏j=1

1

sinhEij

2

, zadjzNanti =

N∏j=1

sinhEij − 2i(γ − γ′)

2. (3.26)

The first part zvzNfund is called zvortex in [78] for 2d U(1) theories (i.e. N = 1). In

this case, the term µi − µj in Eij is simply ignored and γ′ = 0 as I ignore SU(2)L.

Also, as there is only one 1d Young diagram of length k in the U(1) case, the

product of s simply runs over s = 1, 2, · · · , k. Rescaling all chemical potentials as

(3d chemical potentials) = β(2d parameters) (3.27)

4The fermions are superpartners of bosonic zero modes only for vortices preserving 4 SUSY inN =4 theories.

113

and taking β → 0 as the 2d limit, keeping all 2d parameters fixed, one obtains

zvortex =k∏s=1

1

2iγ(k − s+ 1)=

1

k!ℏk, (3.28)

where ℏ ≡ 2iγ, apart from β dependent factor which in our case cancels with

other contributions (and in N = 2 theories like [78] should be absorbed into the

fugacity q for vorticity). This agrees with [78]. The extra part zadjzNanti seems to be

unexplored in the N = 2 context.

For Nf > N , I also have extra contributions from Nf − N hypermultiplets,

which decomposes to Nf −N fundamental and anti-fundamental chiral multiplets.

From our N = 4 formula, the contributions of these two are

zNf−Nfund =

Nf∏p=N+1

1

sinhE′

ip

2

, zNf−Nanti =

Nf∏p=N+1

sinhE′ip + 2i(γ − γ′)

2, (3.29)

where the ‘anti-chiral’ contribution again denotes that from the fermion zero modes

of qp for p = N+1, · · · , Nf . As explained above, I took R = −R. To compare these

with 2d N = 2 results, I take γ′ = 0. Let us identify the Scherk-Schwarz masses

of the fields qi, q†i from the chemical potentials, as I reduce the theory to 2d. The

masses are proportional to

qi : µi(1) + µp(−1) + 2iγR+ 2iγ′R = µip + 2iR(γ + γ′)

(qi)† : µi(1) + µp(−1) + 2iγ(−R) + 2iγ′(−R) = µip − 2iR(γ + γ′) . (3.30)

Taking −12 times these to be the masses, the fundamental and the anti-fundamental

chiral multiplets have the following difference in their masses:

mq = mq + 2iR(γ + γ′) → mq + i(γ + γ′) . (3.31)

114

In particular, in the U(1) case (N = 1), one finds (with γ′ = 0)

zNf−Nfund =

k∏s=1

1µip2 − 2iγ(s− 1)

=k∏s=1

1

−mq − 2iγ(s− 1)− iγ/2(3.32)

zNf−Nanti =

k∏s=1

(µip2

− 2iγ(s− 1) + iγ)

=

k∏s=1

(−mq − 2iγ(s− 1) + 3iγ/2) . (3.33)

The analogous N =2 result of [78] is5

zfund =k∏s=1

1

m+ (s− 1)ℏ, zanti =

k−1∏s=0

(m+ (s− 1)ℏ) (3.34)

for a given mass m for a chiral or anti-chiral mode. Up to an overall shift of our

masses by −iγ/2, this is same as our result, up to factors of −1 which in our case

all cancel out in (3.21).

It would be illustrative to take a more detailed look at the formula for single

vortices, to explain the index interpretation in some cases and also to emphasize a

subtlety. From (3.20), one has N different saddle points. The total index at single

vorticity is thus given by their sum:

Ik=1 =sin(γ + γ′)

sin 2γ

N∑i=1

N∏j( =i)

sinh(µji+2i(γ−γ′)

2

)sinh

µji2

Nf∏p=N+1

sinh(µpi−2i(γ−γ′)

2

)sinh

µpi2

. (3.35)

From the calculation of 1-loop determinants at k = 1 in appendix C, one can easily

show that the factor sin(γ+γ′)sin 2γ combines the contribution sin−2(2γ) from the center-

of-mass zero modes and sin 2γ sin(γ + γ′) from the Goldstone fermion zero modes

for 4 broken supercharges. γ, γ′ are lifting, or regularizing, these zero modes. So I

5I corrected the ranges of s summation in eqns.(2.29) and (2.30) of [78], based on theireqns.(2,23), (2.27) and (2.28). In any case, this difference can be absorbed by an overall shiftof all masses by ℏ.

115

shall call them as the center-of-mass index

Icom =sin(γ + γ′)

sin 2γ(3.36)

for a single super-particle. To get the real information on bound state degeneracies,

one has to expand the denominator in certain powers of the fugacity eiγ and extract

out their (integral) coefficients. Just like the instanton index in 5d theories studied

in [32], expanding (3.36) in the fugacity is ambiguous. However, just as in [32], it

suffices to identify a factor of (3.36) as accompanying the translation degree per

super-particle, factored out from the more informative internal degeneracy factor.

In fact, as I shall explain in more detail in the next section, one always obtains a

single factor of (3.36) when one extracts out the single particle partition function

from the general multi-particle result (3.21). This factor is also ignored in all sorts

of bound state counting with translational zero modes.

The remaining factor in (3.35) is more nontrivial. For the case with Nf = N ,

namely for local vortices, this remainder becomes very simple after one sums over

N saddle points. With explicitly summing over them for a few low values of N , one

can easily confirm that

Ik=1 = Icom(γ, γ′)(ei(N−1)(γ−γ′) + ei(N−3)(γ−γ′) + · · ·+ e−i(N−1)(γ−γ′)

)≡ Icom χN (γ − γ′) (3.37)

for Nf = N . The number of states χN (0) = N from the internal part of moduli

space is finite, which is simply due to the compactness of the internal moduli space

CPN−1 for local vortices. So, as the trace expression (3.19) obviously implies, the

index for Nf =N can be naturally regarded as a Witten index counting degeneracy.

For semi-local vortices with Nf >N , the remainder of (3.35) is subtler. Unlike

the case with Nf =N , the flavor chemical potentials µi survive even after one sums

over N saddle points. In particular, the dependence on sinhµij2 in the denominator

survives in the index, making its expansion in the fugacities eµi again ambiguous

116

like (3.36). Just like (3.36), the chemical potentials µi regularizes the internal zero

modes, some of them being noncompact for Nf >N . For the index for 5d instanton

particles, two interpretations were provided to such an internal index in different

contexts [32]. Firstly in the Coulomb branch in which U(N) is broken to U(1)N ,

the signs of the Noether charges for U(1)N are fixed in a BPS sector, making the

expansion of the denominator unambiguous. Secondly, in the symmetric phase, in

which the whole U(N) gauge symmetry is unbroken, the same index was proven

to be a superconformal index which counts gauge invariant operators, after a well-

defined U(N) singlet projection. Here for semi-local vortices, it seems that the

vortex partition function is similar to neither of the two cases. As S[U(N)×U(Nf−

N)] is a global symmetry, the gauge invariance projection is unnecessary. Also, since

this symmetry is unbroken, there is no fixed sign for their Cartans either. Rather,

one should expand the expression (3.35) into the irreducible characters of the global

symmetry. I have attempted this expansion of (3.35). It does not clearly work in

an unambiguous way, essentially due to an ambiguity on how to go around the

poles in (3.35). So for semi-local vortices, I do not have a clear understanding of

its Witten index interpretation, despite its formal expression (3.19) as trace over

Hilbert space. Perhaps a new interpretation of noncompact internal modes from a

parton-like picture [79] might be necessary.

Even without a solid index interpretation, I can get useful information from

them, regarded a kind of supersymmetric partition functions on R2 × S1. In the

next section, I use them to study Seiberg dualities. The index interpretation helps

when available, but is not essential.

The index for the N = 3 theory turns out to be very similar to the above

N = 4 index, with small changes. From the quantum mechanics analysis, I obtain

117

the general formula

Ik1,k2,··· ,kN = (−1)ke−S0

N∏i=1

ki∏s=1

N∏j=1

sinhEij−2iγ

2

sinhEij

2

Nf∏p=N+1

sinhE′

ip−2iγ(R+R)+2iγ

2

sinhE′

ip−2iγ(R+R)

2

(3.38)

with same definitions of Eij , E′ip, and

e−S0 = e−κ∑N

i=1

∑kis=1[µi−2iγR−4iγ(s−1)] . (3.39)

Note that the U(1)R charge R does appear in S0 part of the index even after setting

R = −R. I set R = 1/2 for the N = 3 theory.

3.4 Seiberg dualities

3.4.1 N = 4 dualities from vortices

Seiberg dualities for N =4 gauge theories in a naive form are motivated by branes

[18]. Consider the brane configuration on the left side of Fig 7 with ζ = 0, where

NS5′-brane is not displaced relative to the NS5-brane in x7, x8, x9 directions. The

resulting U(N)N = 4 gauge theory is coupled toNf (≥ N) fundamental hypermul-

tiplets. At low energy, this theory may (but not always) flow to a superconformal

field theory. Now consider the configuration obtained by letting the two NS5-branes

to cross each other by moving along x6 direction. By the brane creation effect [55],

on the right side of the figure there are Nf −N D3-branes stretched between two

NS5-branes. So one obtains an N = 4 U(Nf − N) gauge theory coupled to Nf

matters. Supposing that both theories flow to SCFT, one would have obtained a

brane realization of two QFT with same IR fixed point, and thus a Seiberg-like

duality. However, as shown in [51], this happens only under a restrictive condition.

The main method of [51] is to study the R-charges of BPS magnetic monopole op-

erators in the UV theory, and see if they can sensibly saturate the superconformal

BPS bounds for the scale dimensions of local operators in IR. Picking an N = 2

118

R-charge R (given by JR + JL in the notation of our previous section), the BPS

scale dimensions ∆ of chiral monopole operators saturate the bound

∆ ≥ R . (3.40)

From the unitarity bound, the scale dimensions these operators should satisfy ∆ ≥12 . So if any of the monopole operators have R smaller than 1

2 , the QFT cannot

flow to an N = 4 SCFT, at least not in a way that uses the UV SO(4) R-charges as

the superconformal R-charges in IR. [51] refers to this as the absence of ‘standard

IR fixed point.’

Considering the monopole operator with the U(N) GNO chargeH = (n1, n2, · · · , nN ),

with integer entries, one obtains the following R-charge

R =Nf

2

N∑i=1

|ni| −∑i<j

|ni − nj | (3.41)

of the monopole operator. Plugging in H = (1, 0, 0, · · · , 0) charge, one obtains a

simple necessary condition

Nf

2−N + 1 ≥ 1

2→ Nf ≥ 2N − 1 (3.42)

for the existence of a standard fixed point. Indeed, if this condition is satisfied,

there are no violations of the unitarity bound for other monopole operators [51].

Now considering the putative Seiberg-dual pair with same numberNf of flavors

and the ranks of gauge groups being N and Nf −N , respectively, it is difficult to

have both theories in the pair to satisfy the bound (3.42). Such cases are [18]

Nf = 2N when Nf is even, and Nf = 2N −1 or Nf = 2N +1 when Nf is odd. The

first case is self Seiberg-dual, and the next two cases are Seiberg-dual to each other

(with fixed Nf ). So the only possible nontrivial Seiberg duality with standard fixed

point will be between the theory with Nf = 2N −1 and another with same Nf and

rank N − 1.

119

However, even the last Seiberg duality has to be understood with care, because

there exists an operator which saturates the unitarity bound ∆ ≥ 12 when Nf =

2N −1. The operator with scale dimension 12 should correspond to a free field, or a

free twisted hypermultiplet [51]. In particular, the case with Nf = N = 1 belongs

to this case, in which case the naive Seiberg dual has rank Nf − N = 0 that the

former cannot be dual to nothing. Thus, even the theory with Nf = 2N −1 cannot

be Seiberg-dual to its ‘naive dual’ in the simplest sense. The modified proposal is

that the theory with Nf =2N−1 is dual to its naive dual times a decoupled theory

of a free twisted hypermultiplet [18]. As I shall see in detail, the decoupled sector

comes from the Abrikosov-Nielsen-Olesen (ANO) vortices created by the monopole

operator with dimension 12 .

Now let us study these dualities using vortex partition functions.

The Higgs vacua of the N = 4 theory form a hyper-Kahler moduli space.

On a subspace of this vacuum manifold with qi = 0, there exist BPS vortices in

the spectrum. This submanifold is compact and takes the form ofU(Nf )

U(N)×U(Nf−N) .

In particular, there is no moduli space if Nf = N . The ‘naive’ Seiberg dual pair

have the same form of this moduli subspace. In particular, I naturally identify the

U(N)× U(Nf −N) global symmetries acting on the two moduli spaces.

I compare our vortex partition functions for the naive dual pairs, as functions

of µi, µp, γ, γ′, q. In the quantum mechanical models for the two types of vortices,

the FI parameter r appearing in section 2.2 corresponds to the distance between

the two NS5-branes. Exchanging the two NS5-branes corresponds to changing the

sign of r. Thus, the vortex partition function for the ‘naive dual’ theory can be

obtained from the original theory by tracing the effects of this sign change. In

the N = 4 theory, the only change is that the roles of k × N variable q and

the (Nf − N) × k variable q are exchanged. So one is naturally led to compare

vortex/anti-vortex spectra in the dual pair as I explained in the previous section,

as the representations under U(k) are conjugated after q, q are exchanged. Two

vortex partition functions have different saddle points, either labeled by division

of k into N integers in the original theory, or into Nf − N integers in the naive

120

dual. To obtain the partition function of the naive dual from the original one, one

should change the roles of µi and µp, and further flip their signs. The last sign flip

is needed as the variables q/q charged in U(N) and U(Nf −N) change their roles,

making their charges flip signs. This flip can be undone by flipping the signs of

γ, γ′, as the index is manifestly invariant under the sign flips of all µi, µp, γ, γ′.

I first consider the index INf

N (q, µ, γ, γ′) with low values of k, after expanding

it as

INf

N =∞∑k=0

qkINf

N,k(µ, γ, γ′) , (3.43)

where INf

N,0 ≡ 1. At unit vorticity, k = 1, I obtain (with R = −R)

INf

N,1 =sin(γ + γ′)

sin 2γ

N∑i=1

N∏j(=i)

sinh(µji+2i(γ−γ′)

2

)sinh

µji2

Nf∏p=N+1

sinh(µpi−2i(γ−γ′)

2

)sinh

µpi2

(3.44)

for the original partition function, and

INf

Nf−N,1 =sin(γ + γ′)

sin 2γ

Nf∑p=N+1

N∏j=1

sinh(µjp+2i(γ−γ′)

2

)sinh

µjp2

Nf∏q(=p)

sinh(µqp−2i(γ−γ′)

2

)sinh

µqp2

(3.45)

for the ‘dual’ partition function. They apparently take very different forms, as the

first and second are sums over N and Nf−N terms, respectively. After summation,

I find that they are related in a simple manner. For simplicity, let us consider the

case in which Nf ≤ 2N : the other case with Nf ≥ 2N can be obtained from this

by changing the roles of two theories. Then, one finds that

INf

N,1− INf

Nf−N,1 =sin(γ + γ′)

sin 2γχ2N−Nf

(γ−γ′) = Icom(γ, γ′)χ2N−Nf

(γ−γ′) , (3.46)

where

χ2N−Nf(γ − γ′)

=ei(2N−Nf−1)(γ−γ′) + ei(2N−Nf−3)(γ−γ′) + · · ·+ e−i(2N−Nf−1)(γ−γ′) (3.47)

121

is the character for the 2N−Nf dimensional representation of SU(2). By definition,

χ0 = 0. I have checked this expression for many cases, varying N,Nf . Note that,

even if INf

N,1 and INf

Nf−N,1 separately depend on µi, µp, their difference on the right

hand side does not. The result says that the single vortex states in the ‘naive’ dual

pair are actually not the same. Rather, the theory with larger gauge group rank N

(> Nf−N) has more states given by the simple expression on the right hand side of

(3.46). The right hand side could be naturally explained if the excess states appear

in a definite SU(2)L representation and are neutral in JR and JE . In particular,

when Nf = 2N − 1, the above formula (3.46) becomes

I2N−1N,1 − I2N−1

N−1,1 = Icom(γ, γ′) , (3.48)

implying that the excess state at k = 1 is just one more single-particle state. This

can appear if the U(N) theory is dual to the U(N − 1) theory (the naive dual)

times a decoupled twisted hypermultiplet with unit vorticity as suggested in [18].

A more reassuring relation is found at O(q2) order. I find that

INf

N,2− INf

Nf−N,2 =I2N−Nf

(γ, γ′)2 + I2N−Nf(2γ, 2γ′)

2+I2N−Nf

(γ, γ′)INf

Nf−N,1 (3.49)

where

I2N−Nf(γ, γ′) ≡ Icom(γ, γ

′)χ2N−Nf(γ − γ′) . (3.50)

Combined with the k = 1 order results, this suggests that the exact relation between

the two vortex partition functions is

INf

N (q, µ, γ, γ′) = INf

Nf−N (q, µ, γ, γ′) exp

[ ∞∑n=1

1

nI2N−Nf

(nγ, nγ′)qn

]. (3.51)

Expanding both sides up to O(q2), one recovers (3.46) and (3.49). I have also

checked (3.51) at O(q3) for a few low values of Nf , N . At Nf = 2N − 1, the

122

exponential factor on the right hand side becomes

exp

[ ∞∑n=1

1

nIcom(nγ, nγ

′)qn

], (3.52)

which is exactly the multi-particle index one obtains from a free twisted hyper-

multiplet with unit vorticity: the single particle index is given by Icom(γ, γ′)q. This

precisely supports the Seiberg duality of [18]. It is also very natural that this free

field carries unit vorticity, as this decoupled sector is suggested from the existence

of a monopole operator with GNO charge (1, 0, 0, · · · , 0) saturating the unitarity

bound. The above free field states should naturally be regarded as being created

by this monopole operator.

It is also interesting to find that the vortex partition functions of the two ‘naive’

dual pairs are related in a very simple manner for general N ≤ Nf , although not

being completely equal. First of all, let us insert Nf = N to (3.51). Then one

obtains

INN = exp

[ ∞∑n=1

1

nIN (nγ, nγ

′)qn

]= exp

[ ∞∑n=1

1

nIcom(nγ, nγ

′)χN (nγ−nγ′)qn],

(3.53)

as IN0 = 1 from the absence of vortices when the gauge group rank is zero. Thus,

when Nf =N , the index is independent of the flavor fugacities µi and takes the

form of the multiparticle states of unit vortices. From the single particle index

IcomχN (γ − γ′)q in the exponent, one finds N different species of ideal vortex

particles. The above partition function may be implying that the low energy theory

could be a free theory of N twisted hypermultiplets. When N = Nf = 1, one simply

gets a free theory description of the (massless) ANO vortex at low energy.

Secondly, inserting (3.53) back to (3.51), one obtains

INf

N (q, µ, γ, γ′) = INf

Nf−N (q, µ, γ, γ′)I

2N−Nf

2N−Nf(q, γ, γ′) . (3.54)

One may interpret this as implying a novel form of IR duality in which the theory

123

with Nf < 2N is dual to the naive Seiberg dual times a decoupled sector, given

by the U(2N−Nf ) theory with 2N−Nf flavors. There should be very nontrivial

requirements for this to be true. Firstly, as the global flavor symmetry of the naive

duals matches to be U(N) × U(Nf − N) in the UV description, the U(2N−Nf )

flavor symmetry of the latter decoupled factor is not visible. So there should be a

U(2N−Nf ) symmetry enhancement of the theory withNf < 2N in IR, for the above

factorized duality to be true. At the level of vortex partition function, the latter

U(2N −Nf ) flavor symmetry is invisible due to the disappearance of its chemical

potentials in the partition function, as explained in the previous paragraph. The

way how such an IR U(2N−Nf ) symmetry enhancement could appear is suggested

by the vortex partition function itself. As the decoupled factor on the right hand

sides of (3.54) and (3.51) is a multi-particle (or Plethystic) exponential of ideal

vortex particles, appearance of 2N−Nf species of decoupled vortex particles in IR

could provide U(2N−Nf ) enhanced symmetry which rotates them.

Such a generalized duality also makes sense if one considers massless sectors.

The Coulomb branches of both theories (at ζ = 0) have dimension 4N , precisely

after including the decoupled sector to the naive Seiberg dual. By studying the

Coulomb/Higgs moduli spaces, it was already noted in [51] that theories with Nf <

2N have some free vector multiplets (or twisted hypermultiplets) in IR, as complete

Higgsing is impossible. Our finding may be regarded as a concrete characterization

of this observation as a generalized Seiberg duality.

One might think that the vortex partition function is a rather special quantity,

probing the qi = 0 region of the Higgs branch only. As a further support, I also note

that a factorization like (3.54) was observed from the 3-sphere partition function,

briefly mentioned in the conclusion of [18]. Using various relations proved in [18],

one can easily show this factorization as follows. The 3-sphere partition function of

a supersymmetric gauge theory is a function of the FI parameter, which they call

124

η, and the real masses mi (i = 1, 2, · · · , Nf ). [18] obtains

ZNf

N (η,mi)

=

Nf

N

( iNf−1eπη

1 + (−1)Nf−1e2πη

)N e2πiη∑Nj=1mj

N∏j=1

Nf∏k=N+1

2 sinhπ(mj −mk)

m

,

(3.55)

where [ ]m denotes symmetrization with the Nf ! permutations on the Nf mass

parameters. The structure of the formula inside the parenthesis is such that Nf

masses are divided into N and Nf −N groups. Therefore, apart from the factor

(iNf−1eπη

1 + (−1)Nf−1e2πη

)Ne2πiη

∑Nj=1mj , (3.56)

the expression is invariant under replacing N by Nf − N , i.e. going to its naive

125

Seiberg-dual. In particular, one obtains

ZNf

N (η,mi) =

Nf

N

( iNf−1eπη

1 + (−1)Nf−1e2πη

)Ne2πiη

∑Nfj=1mj

×

e−2πiη∑Nf

j=N+1mj

N∏j=1

Nf∏k=N+1

2 sinhπ(mj −mk)

m

=(−1)N(Nf−N)

(iNf−1eπη

1 + (−1)Nf−1e2πη

)N (iNf−1e−πη

1 + (−1)Nf−1e−2πη

)N−Nf

× e2πiη∑Nf

j=1mjZNf−NNf

(−η,mi)

=

(iNf−1e±πη

1 + (−1)Nf−1e±2πη

)2N−Nf

e2πiη∑Nf

j=1mjZNf−NNf

(−η,mi)

=(−1)Nf (Nf−N)

(i(2N−Nf )−1e±πη

1 + (−1)(2N−Nf )−1e±2πη

)2N−Nf

e2πiη∑Nf

j=1mjZNf−NNf

(−η,mi)

=(−1)Nf (Nf−N)Z2N−Nf

2N−Nf(±η;

2N−Nf∑j=1

Mj = ±Nf∑j=1

mj)ZNf−NNf

(−η;mi) .

(3.57)

Thus, apart from the possible −1 sign for odd Nf (Nf −N), the partition function

of the theory with Nf < 2N factorizes into two, to the naive Seiberg-dual partition

function and another one with both N,Nf replaced by 2N −Nf .6

It should be interesting to study this possibility of novel IR fixed points further,

and hopefully to shed more light on possible phases of 3d supersymmetric theories.

I hope the clues provided by the vortex partition function in this chapter and the

3-sphere partition function of [18] could provide guiding information for uncovering

some aspects of this subject. Incidently, [61] studied the N = 2 Seiberg dualities of

[54] in the context of 3-sphere partition function and Z-extremization, and made

a similar observation that IR symmetry enhancement and appearance of a free

sector are needed. Also, studies of enhanced symmetry and novel IR fixed points

6The extra −1 sign also exists for Nf = 2N − 1, omitted in [18]. The iNf−1 factor in (3.55)causes this sign.

126

in 4 dimensions are made recently in [80], using the superconformal indices.

3.4.2 Aspects of N = 3 dualities from vortices

Let us now consider the N = 3 (Yang-Mills) Chern-Simons-matter theories with

U(N)κ gauge group and Nf fundamental hypermultiplets. These theories have

Seiberg duality as discussed in [57]: the above theory is proposed to be dual to the

U(Nf+|κ|−N)−κ theory with Nf hypermultiplets. The duality is proposed to hold

in the range 0 ≤ N ≤ Nf + |κ|. This duality has been studied in quite a detail. The

3-sphere partition function was studied in [18], which proved mostly numerical

agreements between the modulus of the two Seiberg-dual partition functions. In

[63], the superconformal indices of some dual pairs are studied and agreements were

shown for certain low values of N,κ,Nf . In the discussion section, I shall point out

a subtlety in this index comparison for more general values of these parameters,

and suggest a possible resolution. Similar issues for N = 2 Seiberg dualities have

been already addressed in [62], which I also revisit later.

In this subsection, I study the proposed Seiberg-dual pair theories after deform-

ing them by an FI parameter, and also discuss the vortex partition function. As ex-

plained in section 2, the FI deformed Chern-Simons-matter (or Yang-Mills-Chern-

Simons-matter) theory has many different branches of partially Higgsed vacua. The

partially Higgsed vacuum with unbroken U(n) gauge symmetry should be dual to

the vacuum with unbroken U(κ− n) symmetry [55]. So to discuss Seiberg duality,

one inevitably has to understand the vortex spectrum in the (partially) unbroken

phase.

As discussed in section 2, there exist two types of brane/string configurations

carrying nonzero vorticity. First type is the topological vortices given by the D1-

brane stretched between D3-branes corresponding to broken gauge groups and the

flavor D3-branes and/or the 5-brane, as shown in Fig 8. Another possible type

is the fundamental string stretched between the D3-branes corresponding to the

unbroken gauge symmetry and other branes, also shown in Fig 8. Since fundamental

127

strings are charged under the unbroken U(n) or U(κ−n) Chern-Simons gauge field,

nonzero vorticity is induced. This yields non-topological vortices [74, 73].

Let us also consider their BPS masses. D1-brane vortices have masses which are

integer multiples of 2πζ. The masses of fundamental strings are integer multiples

of 2πζκ , as this length is determined by a triangle formed by the (1, κ) brane in Fig

8. So in general, when one compares the spetra of the Seiberg-dual pair in partially

broken phases with generic Chern-Simons level κ > 1, one would have both integral

and fractional vortices.

Here I first comment on the spectra when one of the pair theories is in the

totally Higgsed phase. In the brane picture, I have n = 0 on the left side of Fig

8. Then, all vorticies in this theory are topological, having integer multiples of

2πζ as their masses. On the other hand, the Seiberg-dual theory is in a vacuum

with unbroken U(κ)−κ Chern-Simons gauge symmetry. So one might naively think

that the dual theory would have fractional vortices with massses being multiples

of 2πζκ , invalidating the duality invariance of the spectrum. A possible resolution

goes as follows. The dynamics of U(κ)±κ Chern-Simons gauge fields, or the Yang-

Mills Chern-Simons gauge fields, with N = 2, 3 supersymmetry is supposed to be

very nontrivial. In U(n)κ N = 2, 3 YM-CS theory, integrating out the fermions

in the vector multiplet with mass kg2YM at low energy yields a 1-loop shift to

the SU(n) part of the Chern-Simons level. It shifts as κ → κ − n when κ > 0,

and oppositely when κ < 0 so that the absolute value of the level decreases. The

point n = |κ| is special as the 1-loop corrected level vanishes. Thus, the SU(κ)

part of the theory is confining at low energy [72]. This is because the remaining

gauge dynamics is governed by pure SU(κ) Yang-Mills theory at zero CS level.

As the BPS fundamental strings are in the fundamental representation of SU(κ),

one should only consider those forming gauge singlets in the confining phase. The

only way of making gauge singlets with BPS matters in fundamental representation

is to form SU(κ) baryons using totally antisymmetric tensor. Thus, gauge singlet

non-topological vortices come in κ-multiples of the above fundamental string, with

their masses being multiples of 2πζ.

128

Compared to the topological vortices, the classical and quantum aspects of

non-topological vortices seem to be relatively ill-understood. So what I can do in

generic case is predicting the quantum degeneracy of non-topological vortices via

duality by studying topological ones. However, in a simple case, I can do more

by using various effective treatments of non-topological vortices and compare with

dual topological vortices studied in this chapter. The remaining part of this section

is devoted to this study.

Consider the theory with N = Nf = 1 at CS level κ. (I shall soon restrict to

the case with κ = 1 for detailed studies.) I consider the pure Chern-Simons matter

theory without Yang-Mills term. Turning off q = 0 as before, the classical bosonic

equation of motion is derived from the following reduced action7

L =κ

4πϵµνρAµ∂νAρ − |Dµq|2 −

4π2

κ2|q|2(|q|2 − ζ)2 . (3.58)

The two minima |q| = ζ and q = 0 of the potential correspond to the Higgs phase

and the symmetric phase. BPS equations for both topological/non-topological vor-

tices are given by

(D1 ∓ iD2)q = 0 , D0q ∓2πi

κq(|q|2 − ζ) = 0 . (3.59)

In [66], vortex domain wall was obtained for κ > 0 and upper signs of the BPS

equations:

q = (2ζ)1/2

√e2πx1/κ

1 + e2πx1/κe−2πζi(x0+x2)/κ , A2 = A0 = −π|q|

2

κ. (3.60)

This is a domain wall along the x2 direction located at x1 = 0, which separates

the symmetric phase q = 0 in x1 < 0 and the broken phase q =√ζ in x1 > 0. The

7Normalization differs from that of [66]. Also, the gauge fields there and here are related byAthere = −Ahere, as the covariant derivative there is different from ours, Dµq = (∂µ − iAµ)q.Some of the equations and solutions are also changed below, either due to this difference or justcorrecting typos there.

129

domain wall has the following linear vortex density and monentum density along

x2 direction:

B =

∫dx1 F12 = −2πζ

κ, P =

∫dx1 T01 =

πζ2

κ. (3.61)

Furthermore, as the BPS energy density is given solely by vorticity without having

domain wall tension, it was argued [66] that one can bend this ‘tensionless domain

wall’ to yield more BPS solutions. The conjecture of [66] is that, at least for large

vorticity, the classical solution for non-topological vortices can be approximated by

a droplet of broken phase q = 0 inside the symmetric phase with q = 0, separated

by a thin vortex domain wall of arbitrary shape.8

It could be possible to quantize this system and count the degeneracy explicitly.

In this chapter, leaving the full discussion of this problem as a future work, I shall

reproduce some characteristic aspects of non-topological vortices coming from the

tensionless domain wall picture, using the dual topological vortex index. This would

nontrivially support both Seiberg duality as well as the tensionless domain wall

picture for non-topological vortices.

As the vorticity and tangential linear monentum density is along the curve

of the domain wall, the charges of a closed-loop have the following behaviors. The

vorticity is proportional to the circumference ℓ of the boundary of the broken phase

region,

k = − 1

2π

∮B =

ζℓ

κ. (3.62)

On the other hand, for a closed loop the total momentum cancels to zero while the

angular momentum is proportional to the area A of the broken phase region:

J =

∮x ∧ Pdx = −2PA = −2πζ2

κA . (3.63)

8Of course one could think of bending the wall to have the unbroken phase outside. Quantummechanically, there should be a sense of doing so. However, as this would yield a topological vortexwith quantized classical vorticity, I expect there to be a subtlety in the above argument at theclassical level.

130

I put a minus sign because J is negative for non-topological vortices with q = 0

region outside the wall. This can be easily seen by noting that the unbroken region

is on the left side of the wall in (3.60), and bending the wall to a non-topological

vortex makes a clockwise circulation of the momentum P with J < 0. As the

circumference ℓ of a curve gives an upper bound for the area A of the region

it surrounds by ℓ2 ≥ 4πA, the vorticity k gives an upper bound to the angular

momentum J as

k2 ≥ 2

κ|J | . (3.64)

This is reminiscent of the angular momentum bounds for other familiar 2-charge

systems. For instance, 14 -BPS 2-charge systems which can be realized as wrapped

D0-D4 or F1-momentum states all come with the angular momentum boundQ1Q2 ≥

|J |, where Q1, Q2 are the two charges. As the electric charge Q of a non-topological

vortex is κ times the vorticity, the bound (3.64) may be written as kQ ≥ 2|J |. Just

from the viewpoint of vortices, this upper bound on J is not so obvious, as putting

many vortices together would naturally yield a bound on J which is linear in k.

It is really the collective linear momentum P along the domain wall which creates

much more angular momentum than the naive expectation.

Below, I show this phenomenon at κ=1 by studying the dual topological vor-

tices in the Seiberg-dual theory. The case with κ=1 is much simpler as the dual

vacuum is in the totally broken phase, admitting topological vortices only. The cases

with |κ| > 1 involve non-Abelian vortex dynamics in the Seiberg-duals (whose do-

main wall description is not explored) and a mixture of non-topological/topologica

vortices in a given vacuum.

At κ=1, one has the U(1) theory with Nf =1 hypermultiplet in the unbroken

phase. I take n = 1 on the left side of Fig 8. As the gauge symmetry is unbroken, I

only need to consider non-topological vortices discussed above.9 In the Seiberg-dual

vacuum, on the right side of the figure, the U(1)−1 gauge symmetry is broken that

9In Fig 8 with N=n=1, it may seem that there are no D3-branes for the string to end on. Itis clearer to move D5’s to have it between NS5- and (1, κ)-branes [57]. Then the string can haveone end on the D5-brane.

131

k 1 2 3 4 5 6 7 8 9 10 11 12

−2Jmax 1 5 13 25 41 61 85 113 145 181 221 265

k 13 14 15 16 17 18 19 20 21 22 23 24

−2Jmax 313 365 421 481 545 613 685 761 841 925 1013 1105

Table 8: Maximal values of the angular momentum −(JE + JR/2)

it suffices for us to consider topological vortices. In this case, the duality predicts

the equality of the non-topological vortex spectrum on one side and the topological

vortex spectrum on the other. I shall study the partition function of the latter and

reproduce (3.64) at κ = 1 (up to a subtlety to be explained below). The general

formula (3.38) at κ = 1 applies to the vortices with∫F12 < 0 at r > 0 (i.e. on

the left side of Fig 8). To get the Seiberg dual anti-vortices at r < 0, one again

exchanges µi and µp in the formula, and then put extra minus signs for all µ’s.

More precisely, I am interested in the single particle bound states. I numerically

obtained the following single particle index

Isp =

∞∑k=1

qkIsp,k(µ, γ) , I(q, µ, γ) = exp

[ ∞∑n=1

1

nIsp(q

n, nµ, nγ)

](3.65)

till O(q30). For instance, the few leading terms are given by

Isp =− qeµ1eiγ + q2e2µ1e5iγ + q3(e7iγ − e13µ1

)+ q4e4µ1

(e9iγ − e11iγ − e15iγ + e17iγ − e19iγ + e25iγ

)+ q5e5µ1

(e11iγ − e13iγ + e15iγ − 2e17iγ + 2e19iγ − e21iγ + 2e23iγ − 2e25iγ

+2e27iγ − e29iγ + e31iγ − e33iγ + e35iγ − e41iγ)+ · · · (3.66)

The maximal value of −(JR + 2JE) for given k can be read off by identifying the

term with maximal power in eiγ at O(qk). There are two cases that we explain

separately.

Firstly, the terms with maximal angular momentum all come with degeneracy

132

1, indicating that the shape of the domain wall curve is indeed rigid so that no

degeneracy is generated. The maximal values of −2J ≡ −2(JR + 2JE) that I find

in zsp for k ≤ 30 are given in Table 8. −2Jmax denotes the maximal value of

−(2JR+4JE) that I find from the single particle index, as the exponent multiplying

iγ as e−iγ(2Jmax) in the index. Plotting k-Jmax, one easily finds that I asymptotically

find Jmax ≈ k2. As I do not expect the R-charge JR to scale as a quadrature of

k, I take it as the asymptotic growth of 2JE and find |JE |max ≈ k2

2 , confirming

the property of non-topological vortices. Moreover, it is easy to check the following

exact relation −Jmax = 2k2 − 2k + 1 for k ≤ 30 from Table 8. This clearly shows

that, ignoring the subdominant terms for large k, the upper bound is quadratic in

k with the correct coefficient.

3.5 Discussions

In this chapter, I studied the supersymmetric partition function on R2 × S1 for

topological vortices in 3d N = 4, 3 gauge theories, in which a U(N) gauge field is

coupled to Nf hypermultiplets. The partition function admits a clear index inter-

pretation for the local vortices when Nf = N . The index interpretation is subtler

for the semi-local vortices for Nf > N , due to the non-compact internal zero modes.

Even in the latter case, the zero modes are lifted by the flavor ‘chemical potentials.’

The partition function is used to study 3d Seiberg dualities.

While studying these dualities, it becomes clear that the duality is exchanging

the light (or perhaps massless in the conformal point) vortices, just like the 4d

Seiberg duality exchanging elementary particles and magnetic monopoles, etc. This

emphasizes the importance of studying vortices and their partition functions for a

better understanding of Seiberg dualities, or more generally strongly coupled IR

physics, in 3d.

The vortex partition functions imply that there may be more possible Seiberg

dualities with N = 4 SUSY than those addressed in the literature. Namely, the

Seiberg dualities of UV theories with ‘standard IR fixed points’ were suggested

133

and studied in [51, 18] at Nf = 2N ∓ 1, with a decoupled twisted hypermultiplet

sector. As seen by the vortex partition function (and also by the 3-sphere partition

function as I reviewed), the duality may extend to the whole window 0 ≤ N ≤ Nf ,

in which a UV theory with Nf < 2N is suggested to be dual to the naive Seiberg

dual times a decoupled sector with N,Nf replaced by 2N −Nf . For this duality to

hold, enhanced IR symmetries and decoupled free sectors have to appear. It should

be interesting to study these issues further.

I also found interesting vortex spectrum in N = 3 Chern-Simons-matter the-

ories, but the structures of the vacua and vortex spectrum are much richer so

that more studies are required. I have compared the vortices in the theory with

N = Nf = 1 and κ = ±1, in which I found some nontrivial agreement between the

proposed Seiberg-dual pair.

There are several directions which I think are interesting.

It would first be interesting to have a definite index interpretation for the par-

tition function of semi-local vortices at Nf >N . In [79], a parton-like interpretation

for these vortex size moduli is given. More precisely, they considered lump solitons

in the CPN sigma model, which are related in IR to our vortices. Also, the partons

from electrically charge particles in [79] appear if I mirror dualize the theory I have

been discussing in this chapter. A more challenging problem along this line would

be the interpretation of the index for 5d instantons [32], perhaps with a similar

partonic picture which could shed light to the 6d (2, 0) SCFT in UV.

I would also like to see if the vortex partition function has any relation to

the superconformal index which counts magnetic monopole operators [49]. This is

conceptually well-motivated as monopole operators are basically vortex-creating

operators. Also, since the vortex partition function yields a good function of chem-

ical potentials in the conformal limit ζ → 0, it might be plausible to seek for an

alternative CFT interpretation of this quantity. The expression of the monopole

index in 3d SCFT is given in [49] as an infinite series expansion in the GNO

charges of monopoles. This contains infinitely many terms, which should be more

efficiently written in some cases. (See next paragraph for a related comment.) Try-

134

ing to rewrite it using the vortex partition functions could provide an alternative

expression for the same quantity. See [81] for a related comment.

As a somewhat remotely related subject, I also remark on tests of 3d Seiberg

dualities with N = 2, 3 supersymmetry in the literatures using monopole operators.

In particular, theN = 3 Seiberg dualities of [57] between Chern-Simons-matter the-

ories are considered in detail. Monopole operators in Chern-Simons-matter theories

are more complicated than those without Chern-Simons term, as magnetic fluxes in-

duce nonzero electric charges which should be screened by turning on matter fields.

Spectrum of such monopoles has been studied either by using localization technique

[49] to calculate the index, or by actually constructing semi-classical monopole so-

lutions at large Chern-Simons level [82]. Tests using the monopole index have been

carried out in [63] for some low values of N,Nf , κ. However, if one considers the

spectrum in full generality for arbitrary N,Nf , κ, apparently one seems to find a

problem about R-charges of monopoles similar to the N = 4 monopoles of [51].

More concretely, the index measures the charge R+2j3 with a chemical potential,

where j3 is the angular momentum of operators on R3. This plays a role analogous

to the R-charge in the index. The lowest value of this charge for a given GNO

charge H = (n1, n2, · · · , nN ) can be obtained from the index, which is

R+ 2j3 =Nf

2

N∑i=1

|ni| −∑i<j

|ni − nj |+ |κ|N∑i=1

(|ni|2

+ n2i

). (3.67)

Although the index only measures R+2j3 charges, inN =3 theories I can separately

say what the values of R and j3 are. This is because they are Cartans of SU(2)R

and spatial SO(3) rotations, which are both non-Abelian. As non-Abelian charges

are not renormalized along continuous deformation of the theory, one can trust

the values of R and j3 obtained from the deformed theory. Similar calculation of

non-Abelian R-charges was explained in [83]. Using this property, one obtains

R =Nf + |κ|

2

N∑i=1

|ni| −∑i<j

|ni − nj | , j3 =|κ|2

N∑i=1

n2i . (3.68)

135

In particular, the expression for the R-charge as seen by the index takes the same

form as the R-charges (3.41) of N =4 monopoles, after replacing Nf by Nf + |κ|.

As the rank bound suggested for the N =3 theory is N ≤ Nf + |κ|, Nf + |κ| plays

the role of Nf in many places. In particular, if N becomes close to Nf + |κ|, one

would have a similar problem of having R-charges, or even R+ 2j3, becoming too

negative.

Practically, the expression for the index in [49] becomes of little use in some

cases. Introducing the fugacity x for R+2j3, the index is given as an expansion with

x < 1 for given GNO charge. However, in various theories with N close to Nf + |κ|,

I find the following problem. The minimal value (3.67) of R+2j3 becomes negative

for some GNO charge. Once I find a negative charge, one can find more monopoles

such that R + 2j3 is unbound from below. This implies that an expansion in x is

ill-defined as one sums over all possible GNO charges. In fact, terms with negative

powers in x should be forbidden for superconformal theories. This problem arises

only at the strongly coupled point in which the ’t Hooft coupling N/k is not small.

The only way this pathological behavior can be eliminated from the index,

if I indeed have SCFT for all N in the range 0 ≤ N ≤ Nf + |κ|, seems to be

that the above terms with negative powers in x all cancel out with other monopole

contributions. Possible cancelations of some monopole contributions to the index

for N = 2 theories of [54] were discussed in [62]. This problem emphasizes the need

for a more efficient expression for the index than those presented in [49], perhaps

using the vortex partition function. Also, seeking for a 3d analogue of the recent

study of the ‘diverging’ 4d superconformal index [80] could be interesting.

Finally, it will be interesting to understand the vortex partition function of

the mass-deformed ABJM theory [84] and learn more about the quantum aspects

of this system as well as its gravity dual. Some works in this direction have been

done in [?, ?, 85]. In particular, the Witten index for the vacua was calculated in

[?], both from QFT and its gravity dual, fully agreeing with each other. However,

the gauge/gravity duality of the vortex spectrum poses a puzzle [?] at the moment.

136

Chapter A

N = (4, 4) gauge theory of IIB strings

The Lagrangian of the 2d N = (4, 4) gauge theory for IIB strings is given by

L = L1 + L2 . (A.1)

L1 is given by

L1 =1

g2QMTr

[−1

4(Fµν)

2 − 1

2(DµφaA)(D

µφAa)− 1

2(Dµaαβ)(D

µaβα) +1

2[aαβ, φaA]

2

+i

2(λαa )

†(Dt +Ds)λαa +

i

2(λAα)†(Dt −Ds)λ

Aα +i

2(λAα )

†(Dt +Ds)λAα +

1

2DIDI

+i

2(λaα)

†(Dt −Ds)λaα −DI(qαqβ(τ

I)αβ +1

2(τ I)αβ[a

βα, aαα]− ζI)+

1

2DI′DI′

−DI′(12(τ I

′)AB[φ

Ba, φaA])− i√

2(λaα)

†[aαβ, λβa ]−

i√2(λAα )

†[aαβ, λAβ]

+i√2(λAα)†[aαβ, λAβ ] +

i√2(λαa )

†[aαβ, λaβ] +i√2(λαa )

†[φaA, λAα]

+i√2(λAα)†[φAa, λαa ]−

i√2(λaα)

†[φaA, λAα ]−

i√2(λAα )

†[φAa, λaα]

]. (A.2)

L2 is given by,

L2 =Tr[−Dµq

αDµqα − φaAqαqαφ

Aa + i(ψa)†(Dt −Ds)ψa + i(ψA)†(Dt +Ds)ψ

A

+√2(ψa)

†(ψAφaA) +√2(ψA)†(ψaφ

Aa) + i√2(λαa )

†qαψa + i√2(λAα)†qαψA

−i√2(ψa)

†qαλαa − i

√2(ψA)†qαλ

Aα)]

. (A.3)

In the Higgs branch, the theory describes IIB strings bound to the NS5-branes,

whose target space is the k instanton moduli space.

137

The reality condition of the scalar fields is given by

aαα =1√2(σm)ααam , aαα =

1√2(σm)ααam , aαα = ϵαβϵαβaββ = (aαα)

† , (A.4)

φaA =1√2(σI)aAφI , φAa =

1√2(σI)AaφI , φAa = ϵabϵABφbB = (φbB)

† , (A.5)

with m = 1, 2, 3, 4 and I = 1, 2, 3, 4. The fermions satisfy the following reality

conditions,

λaα = −ϵαβϵab(λbβ)† , λAα = ϵαβϵAB(λBβ )

† , (A.6)

λαa = −ϵαβϵab(λβb )† , λAα = ϵαβϵAB(λBβ)† . (A.7)

138

Chapter B

SUSY and cohomological formulation

for vortex quantum mechanics

In this appendix I construct the cohomological formulation of the vortex quantum

mechanics which is useful in the Witten index computation. The quantum mechan-

ical model for N = 4 vortices was introduced in section 2.2. The lagrangian (3.16)

preserves 4 real supersymmetries Qa, which are given by

QaAt= iλa , QaAt = −iλa

QaϕI = i(τ I) ba λb , Qaϕ

I = i(τ I) ba λb

Qaλb = (τ I)ab

(−Dtϕ

I +1

2ϵIJK [ϕJ , ϕK ]

)+ iϵabD

Qaλb = (τ I)ab

(Dtϕ

I +1

2ϵIJK [ϕJ , ϕK ]

)− iϵabD (B.1)

for the vector multiplet,

QaZ =√2χa , QaZ

† = −√2χa

Qaχb = −i√2ϵabDtZ −

√2(τ I)ab[ϕ

I , Z]

Qaχb = −i√2ϵabDtZ

† +√2(τ I)ab[ϕ

I , Z†] (B.2)

for the adjoint chiral multiplet, and

Qaq =√2ψa , Qaq

† = −√2ψa

Qaψb = −i√2ϵabDtq −

√2(τ I)abϕIq

Qaψb = −i√2ϵabDtq

† −√2(τ I)abq

†ϕI (B.3)

139

for the N fundamental chiral multiplets. Similarly, one can obtain the SUSY trans-

formations of Nf −N anti-fundamental chiral multiplets q, ψ. For N = 3 vorticies,

the lagrangian differs by the term (3.18) and only 2 real supercharges Q2 (and its

complex conjugation) are preserved.

To define the Witten index (3.19), I choose one supercharge among Qa

Q ≡ 1√2(Q2 + Q1) . (B.4)

The index counts the BPS particles annihilated by Q. I can develop a cohomological

formulation using Q. Let us define

ϕ ≡ At + ϕ3 , ϕ ≡ −At + ϕ3 , ϕ± ≡ 1√2(ϕ1 ± iϕ2)

η ≡√2i(λ1 − λ2) , Ψ ≡ i√

2(λ1 + λ2) . (B.5)

The lagrangian and theQ transformation can be rewritten with these new variables.

The Q transformation for the vector multiplet is given by

Qϕ = 0 , Qϕ = η , Qη = [ϕ, ϕ]

Qϕ+ = iλ2 , Qϕ− = iλ1 , Q2ϕ± = [ϕ, ϕ±]

QΨ = E ≡ −i[ϕ1, ϕ2]−D , Q2Ψ = [ϕ,Ψ] , (B.6)

where I omitted time derivatives ∂t acting on the fields as I use a ‘matrix model’

like notation for convenience. One can restore time derivatives by replacing −iAtby Dt whenever I need. I see that the square of the supercharge, Q2, acting on

the fields yields the gauge transformation generated by the complexified parameter

ϕ. Accordingly, Q is nilpotent operator on-shell (I used the fermion equation of

motion for the last equality in (B.6)) up to the gauge rotation. This fact allows us

to construct Q cohomology. Since off-shell nilpotency is required for localization, I

140

introduce an auxiliary scalar H and modify the supersymmetry transformation as

QΨ = H , QH = [ϕ,Ψ] . (B.7)

To have the off-shell invariant action the bosonic potential should also be changed

as follows

−1

2tr(E2)

→ tr

(1

2H2 −HE

). (B.8)

By integrating out H, I can recover the original action and the supersymmetry.

It is straightforward to generalize the cohomological formulation to the chiral

multiplets. I obtain the supersymmetry transformation

QZ = χ2, Qχ2 = [ϕ,Z], Qχ1 = FZ ≡ −[ϕ1 − iϕ2, Z], Q2χ1 = [ϕ, χ1], (B.9)

for the adjoint chiral multiplet,

Qq = ψ2 , Qψ2 = ϕq , Qψ1 = Fq ≡ −(ϕ1 − iϕ2)q , Q2ψ1 = ϕψ1

Qq = ψ2 , Qψ2 = −qϕ , Qψ1 = Fq ≡ q(ϕ1 − iϕ2) , Q2ψ1 = −ψ1ϕ , (B.10)

for the N fundamental and for the Nf − N anti-fundamental chiral multiplets,

respectively. I again introduce auxiliary scalars hZ , hq, hq and find the following

off-shell supersymmetry

Qχ1 = hZ , QhZ = [ϕ, χ1]

Qψ1 = hq , Qhq = ϕψ1

Qψ1 = hq , Qhq = −ψ1ϕ . (B.11)

The bosonic potential containing these auxiliary scalars is written as

tr(hZh

†Z + hqh

†q + hqh

†q − (FZh†Z + Fqh†q + Fqh†q + c.c)

). (B.12)

141

Collecting all the results, the bosonic part of the Euclidean lagrangian can be

written as

LB = tr

(1

8[ϕ, ϕ]2 − 1

2[ϕ, ϕI ][ϕ, ϕI ]− 1

4|[ϕ− ϕ, Z]|2 + 1

4|[ϕ+ ϕ, Z]|2 − 1

2H2 +HE

+hZh†Z + hqh

†q + hqh

†q − (FZh†Z + Fqh†q + Fqh†q + c.c)

)(B.13)

where I = 1, 2.

B.1 Saddle points

I evaluate the Witten index (3.19) using localization. The index can be represented

by a path integral, using the lagrangian with Euclidean time τ (t ≡ −iτ). The time

τ is now periodic with periodicity β and the dynamical variables satisfy periodic

boundary conditions due to the insertion of (−1)F to the index, which makes the

path integral to be supersymmetric. Indeed, the Hamiltonian of the Witten index

is the square of the supercharge Q2 = H. This implies that the Witten index does

not depend on the parameter β.

I also introduce chemical potentials µi, γ and γ′ to the path integral. The

boundary conditions of the fields are twisted by these chemical potentials. There

is an alternative way to deal with this twisting using the twisted time derivative.

Under the twisting the time derivative is shifted as

Dτ → Dτ −µi

βΠi − i

γ

β(2J)− i

γ′

β(2J ′) . (B.14)

Note that J and J ′ commute with the supercharge Q, so the deformed lagrangian

is still invariant under Q. See [32] for details.

The index is independent of the continuous parameters β, r. So I can take

any convenient values of these parameters for the calculation. I consider the limit

β → 0, r → ∞, after which the path integral is localized around the supersymmetric

saddle points. At the saddle point, all fermionic fields are set to zero and the bosonic

142

fields are constrained by supersymmetry:

Qη = [ϕ, ϕ] = 0 , Qψ1 = Fq = 0 , Qψ1 = Fq = 0 , Qχ1 = FZ = 0

Qψ2 = ϕq − q µβ + 2i(γJ+γ′J ′)β q = 0 , Qψ2 = −qϕ+ µ

β q +2i(γJ+γ′J ′)

β q = 0

Qχ2 = [ϕ,Z] + 2i(γJ+γ′J ′)β Z = 0 ,

QΨ = −i[ϕ1, ϕ2]−([Z,Z†] + qq† − q†q − r

)= 0 . (B.15)

I integrated out all the auxiliary scalars H,hZ , hq, hq, and the chemical potential µ

here is a diagonal N ×N matrix. The last three equations on the first line can be

solved by setting ϕ1− iϕ2 = 0. Using the U(k) gauge transformation and [ϕ, ϕ] = 0

condition, I can take the saddle point value of ϕ to be a diagonal k × k matrix. ϕ

also becomes a diagonal k × k matrix at the saddle point, but the exact value is

not determined by the above equations. It will be determined later by using the

equation of motion of ϕ. The remaining equations reduce to

ϕqi − µi − 2i(γ + γ′)Rqβ

qi = 0 , qpϕ− µp + 2i(γ + γ′)Rqβ

qp = 0 ,

[ϕ,Z] +4iγ

βZ = 0 , [Z,Z†] + qiq†i − qp†qp = r . (B.16)

These equations imply that the full solutions can be constructed by using the

notion of k dimensional vector space. The k × k matrices ϕ,Z act as operators

on the vector space and qi and qp† can be regarded as Nf eigenvectors of ϕ with

eigenvalues µi−2i(γ+γ′)Rβ and

µp+2i(γ+γ′)Rβ , respectively. Two eigenvectors qi and

qp† are orthogonal to each other unless they are the same type, namely qpqi = 0,

q†i qj = 0 for i = j and qpq

r† = 0 for p = r. Considering an eigenstate |λ⟩ with ϕ|λ⟩ =

λ|λ⟩ and acting the operator Z on it, one can obtain the other state with shifted

eigenvalue, Z|λ⟩ = |λ− 4iγβ ⟩. Therefore, Z behaves as the raising operator shifting

the eigenvalue of the states by −4iγβ and its conjugate acts as the lowering operator

shifting the eigenvalue in opposite way. It is possible to obtain the complete basis

of the k dimensional vector space from the ground state, defined to be annihilated

143

by Z†, by acting Z many times.

For r > 0, I find q = 0 from the last equation of (B.16). The same phenomenon

happens to the instanton calculus and the proof is given in [32]. A similar argument

holds in our case and I can set q to zero at the generic saddle point. Thus I only

consider the eigenstates qi and their descendants

|m⟩i ∝ Zmqi , (B.17)

with eigenvalues µ−2i(γ+γ′)R−4imγβ of ϕ for m ≥ 0 and i = 1, 2, · · · , N . As the vector

space is finite dimensional, it will terminate at some number m. There is a 1-to-1

correspondence between the set of these eigenstates and one dimensional Young

diagram. The number of the boxes in the Young diagram is determined by the

number of states in the corresponding set. There are N such Young diagrams and,

since the vector space is k dimensional, the total number of boxes in the Young

diagrams should be k. Therefore, the saddle points can be classified by the one

dimensional N -colored Young diagrams with total box number k. For the given

colored Young diagrams, the explicit values of fields of the corresponding saddle

point are determined by solving the last equation of (B.16).

As an example, let us find the saddle point solutions for some low values of k,

using the above construction. I first consider the case with k = 1. Here qi is simply a

number for each i. Only one of N numbers, say i’th one, can be nonzero. The vector

qi has eigenvalue µi−2i(γ+γ)Rβ of ϕ. It is annihilated by Z as the total vector space

is k = 1 dimensional, so Z = 0. The last equation of (B.16) fixes qi =√reiθ where

θ is the phase for the broken U(1) on the i’th D3-brane. The phase factor θ can

be eliminated by the unbroken U(1) gauge symmetry of the single vortex quantum

mechanics. I write the i’th saddle point as i from the colored Young diagram

notation. This can be understood as a single D1-brane bound to i’th D3-brane.

At k = 2, there are two kinds of saddle points. The first saddle point takes

( i, j) form of the colored Young diagram. This is a superposition of two k = 1

saddle points where qi, qj (i = j) contain nonzero components. Here, qis are two

144

dimensional vectors. It corresponds to one D1-brane bound to i’th D3-brane and

the other D1-brane bound to j’th D3-brane. Using U(2) gauge transformation I

can write the solution as

qi =√r(1 0) , qj =

√r(0 1) ,

ϕ = diag

(µi − 2i(γ + γ′)R

β,µj − 2i(γ + γ′)R

β

), (B.18)

with Z = 0. This solves all equations in (B.16). I can also consider two phase

factors θ1,2 for qi, qj corresponding to the unbroken U(1)2 gauge symmetry on two

D3-branes, but they can be eliminated by U(1)2 ⊂ U(2) gauge transformation of

two vortices.

The second saddle point is given by the colored Young diagram i. In this

case, only one vector qi among N vectors has nonzero component. I can write it

as qi = λ|1⟩ where ϕ|1⟩ = µi−2i(γ+γ′)Rβ |1⟩. I need one more state to form a two

dimensional vector space. It will be obtained by acting Z on |1⟩ once. Thus I find

|2⟩ ∝ Z|1⟩ , Z = c|2⟩⟨1| , (B.19)

which implies that 2-1 component of 2 × 2 matrix Z gets nonzero value c. The

state |2⟩ is killed by Z so there is no more state. Two eigenstates |1⟩, |2⟩ form a

complete basis of k = 2 dimensional vector space. The last equation of (B.16) again

fixes the undetermined constants λ, c and, using the U(2) gauge transformation,

the solution is given by

qi=√2r(1 0) , Z=

0 0√r 0

, (B.20)

ϕ=diag

(µi − 2i(γ + γ′)R

β,µi − 2i(γ + γ′)R− 4iγ

β

). (B.21)

This solution illustrates two vortices bound to the single i’th D3-brane.

Finally, I explain the solution for ϕ. The saddle point value of ϕ is not fully

145

determined by the equation (B.16). It only imposes the condition [ϕ, ϕ] = 0 which

can be solved by taking a diagonal ϕ. I should use the equation of motion of ϕ

(which is a Gauss’ law constraint for the U(k) gauge symmetry) to determine the

saddle point value of ϕ. The variation δϕ yields

−[Z†, [ϕ, Z]− 4iγ

βZ

]+

1

2

(ϕqq† + qq†ϕ

)+ q

µ− 2i(γ + γ′)R

βq† = 0 , (B.22)

with q and Z taking the saddle point values. This equation is easily solved by

setting ϕ = −ϕ. One can check it using the first three equations in (B.16).

B.2 Determinants

I now compute the 1-loop determinant of the path integral around the saddle points

obtained above. I localize the path integral by taking the limit β → 0, r → ∞ since

the Witten index is independent of these parameters. Then the 1-loop determinant

of the quadratic fluctuations with the classical action will give the exact result in

this limit. The quadratic terms of the bosonic fields around a generic saddle point

is given by

L(2)B =

1

8

(δϕ+ δ ˙ϕ− [δϕ, ϕ]− [ϕ, δϕ]

)2+ |[δϕI , Z]|2 + qq†δϕIδϕI

+1

4|[ϕ+ ϕ, δZ] + [δϕ+ δϕ, Z]|2 + 1

4|(ϕ+ ϕ)δq + (δϕ+ δϕ)q|2 + 1

4|δq(ϕ+ ϕ)|2

+1

2

(δϕI − [ϕ, δϕI ]−

2i(γJ + γ′J ′)

βδϕI

)(δϕI + [ϕ, δϕI ]−

2i(γJ + γ′J ′)

βδϕI

)+

(δZ† +

1

2[δZ†, ϕ− ϕ] +

1

2[Z†, δϕ− δϕ] +

4iγ

βδZ†

)×(δZ − 1

2[ϕ− ϕ, δZ]− 1

2[δϕ− δϕ, Z]− 4iγ

βδZ

)+

(δq† +

1

2δq†(ϕ− ϕ) +

1

2q†(δϕ− δϕ)− µ

βδq† +

2i(γ + γ′)Rqβ

δq†)

×(δq − 1

2(ϕ− ϕ)δq − 1

2(δϕ− δϕ)q + δq

µ

β− 2i(γ + γ′)Rq

βδqi

)

146

+

(δ ˙q† − 1

2(ϕ− ϕ)δq† + δq†

µ

β+

2i(γ + γ′)Rqβ

δq†)

×(δ ˙q +

1

2δq(ϕ− ϕ)− µ

βδq − 2i(γ + γ′)Rq

βδq

)+

1

2

(δqq† + qδq† + [δZ, Z†] + [Z, δZ†]

)2, (B.23)

where I = 1, 2 and I used the facts ϕI = 0, q = 0 at the saddle points. Note

that all the coefficient are quadratures of γβ ,

γ′

β , ∂τ and the saddle point values of

the fields ϕ,Z, q. Here ∂t ∼ 1β because the time direction is compactified with

the radius β. Also, the bosonic fields take the saddle point values proportional to

1β or

√r. Therefore, when β → 0, r → ∞, the above quadratic terms dominates

other higher order terms so that the saddle point approximation can be applied.

Similar argument reduces the fermionic action to the quadratic action around the

saddle points. So I can obtain the exact value of the Witten index of the vortex

moduli space by evaluating the 1-loop integral of the bosonic terms (B.23) and the

fermionic quadratic terms given by

L(2)F =− λ1

(λ1 + [ϕ, λ1]− 2i(γ − γ′)

βλ1)− λ2

(λ2 − [ϕ, λ2]

)− χ1

(χ1 + [ϕ, χ1]− 4iγ

βχ1

)− χ2

(χ2 − [ϕ, χ2]− 2i(γ + γ′)

βχ2

)− ψ1

(ψ1 + ϕψ1 + ψ1µ

β− 2i(γ + γ′)Rq

βψ1

)− ¯ψ1

(˙ψ1 − ψ1ϕ− µ

βψ1 − 2i(γ + γ′)Rq

βψ1

)− ψ2

(ψ2 − ϕψ2 + ψ2µ

β− 2i(γ + γ′)Rq − 2i(γ − γ′)

βψ2

)− ¯ψ2

(˙ψ2 + ψ2ϕ− µ

βψ2 − 2i(γ + γ′)Rq − 2i(γ − γ′)

βψ2

)+√2i(χa[λa, Z] + [Z†, λa]χa + qψaλa + λaψaq

†)

(B.24)

I will provide the detailed computation of the determinant for one vortex. For

k = 2, I have also performed this calculation which also confirms the general result

147

of section 2.

For k = 1, I can ignore the commutators of the adjoint field, and I can also set

Z to zero from the saddle point analysis, which makes the calculation much easier.

I first consider the bosonic part that gives the following contribution to the index.

1. δZ : The quadratic action of the scalar Z is given by

δZ†(2πin

β+

4iγ

β

)(−2πin

β− 4iγ

β

)δZ ,

for n’th Fourier mode of Z where Z ∼ e− 2πin

βτ. The determinant is given by

[N 2 sin2 2γ

]−1, (B.25)

where N ≡ − 2iβ1/2

∏n =0

(−2πinβ1/2

).

2. δϕ1,2 : The action is given by

δϕ−

[(2πin

β− 2i

γ − γ′

β

)(−2πin

β+ 2i

γ − γ′

β

)− r

]δϕ+ ,

where ϕ± ∼ ϕ1 ± iϕ2. The 1-loop contribution is

[N 2 sin

(γ − γ′ − i

√rβ2

2

)sin

(γ − γ′ + i

√rβ2

2

)]−1

. (B.26)

3. δqp : The action is given by

−δq†p(2πin

β− µi − µp − 2i(γ + γ′)(Rq +Rq)

β

)2

δqp ,

whose one loop determinant is

∏p

[N 2 sinh2

(µi − µp − 2i(γ + γ′)(Rq +Rq)

2

)]−1

. (B.27)

148

4. δqj =i : The action is given by

δq†j

(2πin

β+µi − µjβ

)(−2πin

β− µi − µj

β

)δqj ,

whose determinant is

∏j =i

[N 2 sinh2

(µi − µj

2

)]−1

. (B.28)

5. δϕ, δϕ, qi : The fluctuation of qi is taken to be

qi = eiθ(√

r +δρ√2

).

The action is given by

1

2(δρ)2 + r(δρ)2 + r

(θ − δAτ

)2+

1

2(δϕ3)2 + r(δϕ3)2 .

I choose the gauge θ = 0, then the Faddeev-Popov determinant is simply 1.

I will compute the integral∫[√2rdρ]d(δAτ )d(δϕ

3)]

× exp

[−∫dτ

(1

2(δρ)2 + r(δρ)2 + r (δAτ )

2 +1

2(δϕ3)2 + r(δϕ3)2

)],

and it gives the result

[N 2 sinh

√rβ2

2

]−1

. (B.29)

I can also compute the fermionic determinants from the fermion quadratic terms.

The results are as follows.

149

1. χa : The determinant is given by

N 2 sin 2γ sin(γ + γ′) . (B.30)

2. ψap : The determinant is given by

N 2∏p

sinh

(µi − µp − 2i(γ + γ′)(Rq +Rq)

2

)× sinh

(µi − µp − 2i(γ + γ′)(Rq +Rq) + 2i(γ − γ′)

2

). (B.31)

3. ψaj with j = i : The determinant becomes

N 2∏j =i

sinh

(µj − µi

2

)sinh

(µj − µi + 2i(γ − γ′)

2

). (B.32)

4. λ2, ψi1 : The action is given by

(λ2 ψi1

) −2πinβ i

√2r

−i√2r −2πin

β

λ2

ψi1

,

and the determinant is given by

N 2 sinh2√rβ2

2. (B.33)

5. λ1, ψi2 : The action is given by

(λ1 ψi2

) −2πinβ − 2iγ−γ

′

β −i√2r

i√2r −2πin

β + 2iγ−γ′

β

λ1

ψi2

,

The determinant is given by

N 2 sin

(γ − γ′ − i

√rβ2

2

)sin

(γ − γ′ + i

√rβ2

2

). (B.34)

150

Collecting all the determinants, I can calculate the index corresponding to the

saddle point labeled by i. One can see the cancellation of terms including the pa-

rameters β, r between bosonic and fermionic contributions. The remaining quantity

does not depend on those parameters as we expected. Summing over all N different

saddle points the full index in one vortex sector is given by

Ik=1 =sin(γ + γ′)

sin 2γ

N∑i=1

N∏j =i

sinhµji+2i(γ−γ′)

2

sinhµji2

Nf∏p=N+1

sinhµpi+2i(γ+γ′)(R+R)−2i(γ−γ′)

2

sinhµpi+2i(γ+γ′)(R+R)

2

,

(B.35)

where µij = µi − µj .

151

Bibliography

[1] D. Gaiotto, “N=2 dualities,” JHEP 08 (2012) 034, arXiv:0904.2715

[hep-th].

[2] K. Becker, M. Becker, and J. H. Schwarz, String Theory and M-Theory,

ch. 3. Cambridge University Press, 2007.

[3] A. Sen, “Dyon - monopole bound states, selfdual harmonic forms on the

multi - monopole moduli space, and SL(2,Z) invariance in string theory,”

Phys. Lett. B329 (1994) 217–221, arXiv:hep-th/9402032 [hep-th].

[4] E. Witten, “String theory dynamics in various dimensions,” Nucl.Phys.

B443 (1995) 85–126, arXiv:hep-th/9503124 [hep-th].

[5] P. Horava and E. Witten, “Heterotic and type I string dynamics from

eleven-dimensions,” Nucl.Phys. B460 (1996) 506–524,

arXiv:hep-th/9510209 [hep-th].

[6] E. Witten, “Some comments on string dynamics,” arXiv:hep-th/9507121

[hep-th].

[7] K. S. Narain, “New Heterotic String Theories in Uncompactified Dimensions

¡ 10,” Phys. Lett. B169 (1986) 41.

[8] P. Horava and E. Witten, “Eleven-dimensional supergravity on a manifold

with boundary,” Nucl.Phys. B475 (1996) 94–114, arXiv:hep-th/9603142

[hep-th].

[9] E. Witten, “Dyons of Charge e theta/2 pi,” Phys. Lett. B86 (1979) 283–287.

[10] A. Sen, “Electric magnetic duality in string theory,” Nucl. Phys. B404

(1993) 109–126, arXiv:hep-th/9207053 [hep-th].

[11] C. Montonen and D. I. Olive, “Magnetic Monopoles as Gauge Particles?,”

Phys. Lett. B72 (1977) 117.

152

[12] E. Witten and D. I. Olive, “Supersymmetry Algebras That Include

Topological Charges,” Phys. Lett. B78 (1978) 97.

[13] B. Julia and A. Zee, “Poles with Both Magnetic and Electric Charges in

Nonabelian Gauge Theory,” Phys. Rev. D11 (1975) 2227–2232.

[14] E. Witten, “Constraints on Supersymmetry Breaking,” Nucl. Phys. B202

(1982) 253.

[15] M. F. Atiyah and N. J. Hitchin, “Low-Energy Scattering of Nonabelian

Monopoles,” Phys. Lett. A107 (1985) 21–25.

[16] N. Seiberg, “Electric - magnetic duality in supersymmetric nonAbelian gauge

theories,” Nucl.Phys. B435 (1995) 129–146, arXiv:hep-th/9411149

[hep-th].

[17] T. Banks and A. Zaks, “On the Phase Structure of Vector-Like Gauge

Theories with Massless Fermions,” Nucl. Phys. B196 (1982) 189.

[18] A. Kapustin, B. Willett, and I. Yaakov, “Tests of Seiberg-like Duality in

Three Dimensions,” arXiv:1012.4021 [hep-th].

[19] O. Aharony, A. Hanany, K. A. Intriligator, N. Seiberg, and M. Strassler,

“Aspects of N=2 supersymmetric gauge theories in three-dimensions,”

Nucl.Phys. B499 (1997) 67–99, arXiv:hep-th/9703110 [hep-th].

[20] J. Kim, S. Kim, and K. Lee, “Little strings and T-duality,”

arXiv:1503.07277 [hep-th].

[21] O. Aharony, M. Berkooz, D. Kutasov, and N. Seiberg, “Linear dilatons, NS

five-branes and holography,” JHEP 9810 (1998) 004,

arXiv:hep-th/9808149 [hep-th].

[22] A. Giveon, D. Kutasov, and O. Pelc, “Holography for noncritical

superstrings,” JHEP 9910 (1999) 035, arXiv:hep-th/9907178 [hep-th].

[23] M. Berkooz, M. Rozali, and N. Seiberg, “Matrix description of M theory on

T**4 and T**5,” Phys.Lett. B408 (1997) 105–110, arXiv:hep-th/9704089

[hep-th].

153

[24] N. Seiberg, “New theories in six-dimensions and matrix description of M

theory on T**5 and T**5 / Z(2),” Phys.Lett. B408 (1997) 98–104,

arXiv:hep-th/9705221 [hep-th].

[25] E. Witten, “On the conformal field theory of the Higgs branch,” JHEP 9707

(1997) 003, arXiv:hep-th/9707093 [hep-th].

[26] O. Aharony and M. Berkooz, “IR dynamics of D = 2, N=(4,4) gauge theories

and DLCQ of ’little string theories’,” JHEP 9910 (1999) 030,

arXiv:hep-th/9909101 [hep-th].

[27] A. Giveon and D. Kutasov, “Little string theory in a double scaling limit,”

JHEP 9910 (1999) 034, arXiv:hep-th/9909110 [hep-th].

[28] A. Giveon and D. Kutasov, “Comments on double scaled little string

theory,” JHEP 0001 (2000) 023, arXiv:hep-th/9911039 [hep-th].

[29] N. Seiberg and E. Witten, “The D1 / D5 system and singular CFT,” JHEP

9904 (1999) 017, arXiv:hep-th/9903224 [hep-th].

[30] O. Aharony, M. Berkooz, S. Kachru, N. Seiberg, and E. Silverstein, “Matrix

description of interacting theories in six-dimensions,” Adv. Theor. Math.

Phys. 1 (1998) 148–157, arXiv:hep-th/9707079 [hep-th].

[31] E. Witten, “Elliptic Genera and Quantum Field Theory,”

Commun.Math.Phys. 109 (1987) 525.

[32] H.-C. Kim, S. Kim, E. Koh, K. Lee, and S. Lee, “On instantons as

Kaluza-Klein modes of M5-branes,” JHEP 1112 (2011) 031,

arXiv:1110.2175 [hep-th].

[33] R. Flume and R. Poghossian, “An Algorithm for the microscopic evaluation

of the coefficients of the Seiberg-Witten prepotential,” Int.J.Mod.Phys. A18

(2003) 2541, arXiv:hep-th/0208176 [hep-th].

[34] U. Bruzzo, F. Fucito, J. F. Morales, and A. Tanzini, “Multiinstanton calculus

and equivariant cohomology,” JHEP 0305 (2003) 054,

arXiv:hep-th/0211108 [hep-th].

154

[35] A. Gadde and S. Gukov, “2d Index and Surface operators,” JHEP 1403

(2014) 080, arXiv:1305.0266 [hep-th].

[36] F. Benini, R. Eager, K. Hori, and Y. Tachikawa, “Elliptic genera of

two-dimensional N=2 gauge theories with rank-one gauge groups,”

Lett.Math.Phys. 104 (2014) 465–493, arXiv:1305.0533 [hep-th].

[37] F. Benini, R. Eager, K. Hori, and Y. Tachikawa, “Elliptic Genera of 2d N =

2 Gauge Theories,” Commun.Math.Phys. 333 no. 3, (2015) 1241–1286,

arXiv:1308.4896 [hep-th].

[38] C. Hwang, J. Kim, S. Kim, and J. Park, “General instanton counting and 5d

SCFT,” arXiv:1406.6793 [hep-th].

[39] M. R. Douglas and G. W. Moore, “D-branes, quivers, and ALE instantons,”

arXiv:hep-th/9603167 [hep-th].

[40] B. Haghighat, A. Iqbal, C. Kozcaz, G. Lockhart, and C. Vafa, “M-Strings,”

Commun.Math.Phys. 334 no. 2, (2015) 779–842, arXiv:1305.6322

[hep-th].

[41] B. Haghighat, C. Kozcaz, G. Lockhart, and C. Vafa, “Orbifolds of

M-strings,” Phys.Rev. D89 no. 4, (2014) 046003, arXiv:1310.1185

[hep-th].

[42] D. Tong, “The holographic dual of AdS3 × S3 × S3 × S1,” JHEP 1404

(2014) 193, arXiv:1402.5135 [hep-th].

[43] T. J. Hollowood, A. Iqbal, and C. Vafa, “Matrix models, geometric

engineering and elliptic genera,” JHEP 0803 (2008) 069,

arXiv:hep-th/0310272 [hep-th].

[44] E. Witten, “Phase transitions in M theory and F theory,” Nucl.Phys. B471

(1996) 195–216, arXiv:hep-th/9603150 [hep-th].

[45] A. Klemm, P. Mayr, and C. Vafa, “BPS states of exceptional noncritical

strings,” Nucl.Phys.Proc.Suppl. 58 (1997) 177, arXiv:hep-th/9607139

[hep-th].

155

[46] J. Kim, S. Kim, K. Lee, J. Park, and C. Vafa, “Elliptic Genus of E-strings,”

arXiv:1411.2324 [hep-th].

[47] H.-C. Kim and S. Kim, “M5-branes from gauge theories on the 5-sphere,”

JHEP 1305 (2013) 144, arXiv:1206.6339 [hep-th].

[48] H.-C. Kim, J. Kim, S. Kim, and K. Lee, “Vortices and 3 dimensional

dualities,” arXiv:1204.3895 [hep-th].

[49] S. Kim, “The Complete superconformal index for N=6 Chern-Simons

theory,” Nucl.Phys. B821 (2009) 241–284, arXiv:0903.4172 [hep-th].

[50] O. Aharony, O. Bergman, D. L. Jafferis, and J. Maldacena, “N=6

superconformal Chern-Simons-matter theories, M2-branes and their gravity

duals,” JHEP 0810 (2008) 091, arXiv:0806.1218 [hep-th].

[51] D. Gaiotto and E. Witten, “S-Duality of Boundary Conditions In N=4 Super

Yang-Mills Theory,” Adv.Theor.Math.Phys. 13 (2009) 721,

arXiv:0807.3720 [hep-th].

[52] S. Pasquetti, “Factorisation of N = 2 Theories on the Squashed 3-Sphere,”

JHEP 1204 (2012) 120, arXiv:1111.6905 [hep-th].

[53] C. Hwang, H.-C. Kim, and J. Park, “Factorization of the 3d superconformal

index,” JHEP 1408 (2014) 018, arXiv:1211.6023 [hep-th].

[54] O. Aharony, “IR duality in d = 3 N=2 supersymmetric USp(2N(c)) and

U(N(c)) gauge theories,” Phys.Lett. B404 (1997) 71–76,

arXiv:hep-th/9703215 [hep-th].

[55] A. Hanany and E. Witten, “Type IIB superstrings, BPS monopoles, and

three-dimensional gauge dynamics,” Nucl.Phys. B492 (1997) 152–190,

arXiv:hep-th/9611230 [hep-th].

[56] A. Kapustin, B. Willett, and I. Yaakov, “Exact Results for Wilson Loops in

Superconformal Chern-Simons Theories with Matter,” JHEP 1003 (2010)

089, arXiv:0909.4559 [hep-th].

[57] A. Giveon and D. Kutasov, “Seiberg Duality in Chern-Simons Theory,”

Nucl.Phys. B812 (2009) 1–11, arXiv:0808.0360 [hep-th].

156

[58] V. Niarchos, “Seiberg Duality in Chern-Simons Theories with Fundamental

and Adjoint Matter,” JHEP 0811 (2008) 001, arXiv:0808.2771 [hep-th].

[59] V. Niarchos, “R-charges, Chiral Rings and RG Flows in Supersymmetric

Chern-Simons-Matter Theories,” JHEP 0905 (2009) 054, arXiv:0903.0435

[hep-th].

[60] A. Kapustin, “Seiberg-like duality in three dimensions for orthogonal gauge

groups,” arXiv:1104.0466 [hep-th].

[61] B. Willett and I. Yaakov, “N=2 Dualities and Z Extremization in Three

Dimensions,” arXiv:1104.0487 [hep-th].

[62] D. Bashkirov, “Aharony duality and monopole operators in three

dimensions,” arXiv:1106.4110 [hep-th].

[63] C. Hwang, H. Kim, K.-J. Park, and J. Park, “Index computation for 3d

Chern-Simons matter theory: test of Seiberg-like duality,” JHEP 1109

(2011) 037, arXiv:1107.4942 [hep-th].

[64] C. Hwang, K.-J. Park, and J. Park, “Evidence for Aharony duality for

orthogonal gauge groups,” JHEP 1111 (2011) 011, arXiv:1109.2828

[hep-th].

[65] A. Amariti, D. Forcella, L. Girardello, and A. Mariotti, “3D Seiberg-like

Dualities and M2 Branes,” JHEP 1005 (2010) 025, arXiv:0903.3222

[hep-th].

[66] S. Kim, K.-M. Lee, and H.-U. Yee, “Supertubes in field theories,” Phys.Rev.

D75 (2007) 125011, arXiv:hep-th/0603179 [hep-th].

[67] A. Hanany and D. Tong, “Vortices, instantons and branes,” JHEP 0307

(2003) 037, arXiv:hep-th/0306150 [hep-th].

[68] M. Shifman and A. Yung, “Non-Abelian semilocal strings in N=2

supersymmetric QCD,” Phys.Rev. D73 (2006) 125012,

arXiv:hep-th/0603134 [hep-th].

157

[69] M. Eto, Y. Isozumi, M. Nitta, K. Ohashi, and N. Sakai, “Manifestly

supersymmetric effective Lagrangians on BPS solitons,” Phys.Rev. D73

(2006) 125008, arXiv:hep-th/0602289 [hep-th].

[70] E. Bergshoeff and P. Townsend, “Super D-branes,” Nucl.Phys. B490 (1997)

145–162, arXiv:hep-th/9611173 [hep-th].

[71] Y. Imamura, “Supersymmetries and BPS configurations on Anti-de Sitter

space,” Nucl.Phys. B537 (1999) 184–202, arXiv:hep-th/9807179

[hep-th].

[72] E. Witten, “Supersymmetric index of three-dimensional gauge theory,”

arXiv:hep-th/9903005 [hep-th].

[73] M. Eto and S. B. Gudnason, “Zero-modes of Non-Abelian Solitons in Three

Dimensional Gauge Theories,” J.Phys. A44 (2011) 095401,

arXiv:1009.5429 [hep-th].

[74] R. Jackiw, K.-M. Lee, and E. J. Weinberg, “Selfdual Chern-Simons solitons,”

Phys.Rev. D42 (1990) 3488–3499.

[75] B. Collie, “Dyonic Non-Abelian Vortices,” J.Phys. A42 (2009) 085404,

arXiv:0809.0394 [hep-th].

[76] Y.-b. Kim and K.-M. Lee, “First and second order vortex dynamics,”

Phys.Rev. D66 (2002) 045016, arXiv:hep-th/0204111 [hep-th].

[77] S. Shadchin, “On F-term contribution to effective action,” JHEP 0708

(2007) 052, arXiv:hep-th/0611278 [hep-th].

[78] T. Dimofte, S. Gukov, and L. Hollands, “Vortex Counting and Lagrangian

3-manifolds,” Lett.Math.Phys. 98 (2011) 225–287, arXiv:1006.0977

[hep-th].

[79] B. Collie and D. Tong, “The Partonic Nature of Instantons,” JHEP 0908

(2009) 006, arXiv:0905.2267 [hep-th].

[80] D. Gaiotto and S. S. Razamat, “Exceptional Indices,” JHEP 1205 (2012)

145, arXiv:1203.5517 [hep-th].

158

[81] T. Dimofte, D. Gaiotto, and S. Gukov, “3-Manifolds and 3d Indices,”

Adv.Theor.Math.Phys. 17 (2013) 975–1076, arXiv:1112.5179 [hep-th].

[82] H.-C. Kim and S. Kim, “Semi-classical monopole operators in

Chern-Simons-matter theories,” arXiv:1007.4560 [hep-th].

[83] M. K. Benna, I. R. Klebanov, and T. Klose, “Charges of Monopole

Operators in Chern-Simons Yang-Mills Theory,” JHEP 1001 (2010) 110,

arXiv:0906.3008 [hep-th].

[84] K. Hosomichi, K.-M. Lee, S. Lee, S. Lee, and J. Park, “N=5,6

Superconformal Chern-Simons Theories and M2-branes on Orbifolds,” JHEP

0809 (2008) 002, arXiv:0806.4977 [hep-th].

[85] N. Lambert, H. Nastase, and C. Papageorgakis, “5D Yang-Mills instantons

from ABJM Monopoles,” Phys.Rev. D85 (2012) 066002, arXiv:1111.5619

[hep-th].

159

초 록

이 논문에서는, 물리계의 초대칭성을 이용하여 자세히 분석 가능한 강하게 상호작용

하는 물리 현상과 그에 동반하는 비섭동적인 양면성에 대해 연구한다.

첫 번째 예는 6차원 II 형 끈이론의 비섭동적 5차원 물체인 N55-브레인위의 근본

끈들이 중력과 상호작용하지 않는 영역에서 나타나는 ‘6차원 II형 작은 끈 이론’에 대

한것이다.이논문에서는 II형작은끈들의세계막에서나타나는 2차원초대칭게이지

이론들에 대하여 연구한다. 특히 IIA 형 작은 끈들에 대하여, 낮은 에너지 영역에서의

비섭동적인 물리량을 계산하기 적절한 새로운 2차원 N = (0, 4) 게이지 이론에 대해

조사한다. 그리고 이 2차원 게이지 이론들을 이용하여 IIA / IIB 작은 끈이론들의

비섭동적 초대칭 분배함수인 타원 종수(Elliptic genus)를 계산하고 끈들의 공간상의

원을 감는 모드와 칼루자-클라인 운동량을 뒤바꾸는 T-양면성을 입증한다.

또한 3차원 N = 4 와 N = 3 초대칭 게이지 이론들의 힉스 상(Higgs phase)에

서 양자화된 자기 선속을 가지고 나타나는 보텍스 솔리톤을 연구하고, 그의 비섭동적

초대칭 분배함수를 계산한다. 이 3차원 힉스 상은 이 3차원 초대칭 게이지 이론들의

재규격화군 흐름 상에서의 초대칭 등각 고정점들의 관련 변수 변형과 연관 있으며, 이

논문에서는기존에제안되었던이초대칭등각고정점상에서의사이버그(Seiberg)-양

면성을소용돌이솔리톤의초대칭분배함수를이용하여입증한다.그리고천-사이먼스

상호작용을 포함하고 있는 N = 3 초대칭 게이지 이론에서는 비-위상적 보텍 솔리톤

들의 동역학적 특성을 양면성을 이용하여 확인한다.

주요어 : 양면성, 초대칭성, 끈이론, 작은 끈, 보텍스, 솔리톤

학번 : 2010-23143

160