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Brane Tilings
Masahito Yamazaki (Univ. Tokyo, Hongo)
2008/Jun/18 @ Komaba
Based on review: M. Y., Fortscr. Phys. 56 (2008) 555.(arXiv:0803.4474 [hep-th])
See also K. Kennaway, Int. J. Mod. Phys. A22 (2007) 2977.(arXiv:0706.1660 [hep-th])
Thanks to my collaborators!
Based on joint works with T. Fujimori, K. Ohta,M. Nitta, Y. Imamura,H. Isono, K. Kimura, K. Ueda, N. Sakai
K. Ueda and M.Y., math.AG/0605780, math.AG/0606548,math.AG/0703267
Y. Imamura, H. Isono, K. Kimura and M. Y., Prog. Theor. Phys.,Vol 117. no 5, pp923 (hep-th/0702049)
Y. Imamura, K. Kimura and M. Y., JHEP 0803:058,2008.(arXiv:0801.3528 [hep-th])
T. Fujimori, K. Ohta, M. Nitta, N. Sakai and M. Y., arXiv:0805.1194[hep-th]
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 2 / 49
The questions
Today’s topic: brane tilings (’05-); Introduced by Hanany and collaborators[Hanany-Kennaway, Franco-Hanany-Kennaway-Vegh-Wecht ’05]
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 3 / 49
The questions
Today’s topic: brane tilings (’05-); Introduced by Hanany and collaborators[Hanany-Kennaway, Franco-Hanany-Kennaway-Vegh-Wecht ’05]
Part I: Foundation of brane tilings(What is brane tiling after all? What is its physical meaning?)
Part II: Generalizations and application of brane tilings(Why study brane tilings? What kind of applications/generalizationsdo they have?)
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 3 / 49
Plan
1 Introduction
2 Foundations of brane tilings
3 Generalizations and applications of brane tilingsApplicationsGeneralizations
4 Summary
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 4 / 49
Part I: What is a brane tiling?
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 5 / 49
Part I: What is a brane tiling?
The answer by Hanany and collaborators: brane tiling is a bipartitegraph on T
2:
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 5 / 49
Part I: What is a brane tiling?
The answer by Hanany and collaborators: brane tiling is a bipartitegraph on T
2:
Powerful technique to study 4d N = 1 supersymmetric quiver gaugetheories corresponding to arbitrary toric Calabi-Yau.
Successfuly applied to AdS/CFT [Franco et. al. ’06]
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 5 / 49
What is brane tiling?
Our answer: [Imamura-Isono-Kimura-M.Y. ’07]
A brane tiling is a physical brane systems consisting of D5-branes andNS5-brane. The bipartite graphs are just convenient ways of representingthem!
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 6 / 49
What is brane tiling?
Our answer: [Imamura-Isono-Kimura-M.Y. ’07]
A brane tiling is a physical brane systems consisting of D5-branes andNS5-brane. The bipartite graphs are just convenient ways of representingthem!
0 1 2 3 4 5 6 7 8 9
D5 ◦ ◦ ◦ ◦ ◦ ◦NS5 ◦ ◦ ◦ ◦ Σ (2-dim surface)
Since 5,7-directions are compactified, the theory on D5-branesreduces to 4d gauge theory.
Due to the presence of NS5-brane, supersymmetry is broken toN = 1.
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 6 / 49
The D5/NS5 brane configuration
Eg. (conifold [Klebanov-Witten ’98]):
0 1 2 3 4 5 6 7 8 9
D5 ◦ ◦ ◦ ◦ ◦ ◦NS5 ◦ ◦ ◦ ◦ ◦ ◦
NS5 ◦ ◦ ◦ ◦ ◦ ◦
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 7 / 49
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 8 / 49
Due to the conservation of NS5-charge, T2 is divided into many regions,
each of which corresponds to either (N, 0), (N, 1) or (N, −1)-branes
NS5 cylinders now merge into one!
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 9 / 49
terminology: division of T2 by cycles of NS5-branes are called the
fivebrane diagram.
When (N, k)-brane with |k| ≥ 2 appear, the gauge theory meaning is notknown, and we restrict ourselves to case with k = 0, ±1.
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 10 / 49
The gauge theory on D5-brane is a quiver gauge theory
SU(N) gauge group (vertex of quiver) is only on (N, 0)-branes, andnot on (N, ±1)-branes
bifundamental fields are at the intersection of (N, 0)-branes (masslessopen string)
1
12
1
22
(brane tiling) (quiver)
A1
B1 B2
A2
A1
B1
B2
A2
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 11 / 49
Due to the presence of NS5-branes, we have a chiral spectrum:
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 12 / 49
A term in the superpotential corresponds to a region of (N, 1)-braneand (N, −1)-brane.
(a) with (N, 1)-brane contributes operators of the form tr(X1X2X3) tothe superpotential, and (N, −1)-brane as in (b) contributes tr(X1X3X2)term.
W = +tr(A1B1A2B2)−tr(A1B2A2B1)
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 13 / 49
The relation with bipartite graphs and quiver diagrams
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 14 / 49
Summary of correspondence
fivebrane bipartite graph quiver diagram quiver gauge theory
(N, 1)-brane white vertex (face) superpotential term(light gray) (+ sign)
(N, −1)-brane black vertex (face) superpotential term(dark gray) (− sign)
(N, 0)-brane face vertex gauge group
open string edge edge bifundamental
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 15 / 49
Relation with toric Calabi-Yau cones
The D5/NS5 system represented by brane tilings are equivalent to thegeometry of toric Calabi-Yau (+ D3-brane) by T-duality along T
2(57).
D5-brane → D3-braneNS5-brane → toric Calabi-Yau
Sasaki−Einstein
D3−branetransverse to singularCalabi−Yau
Tip of Calabi−Yau cone
Calabi−Yau Cone
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 16 / 49
Rule
write NS5-cycle with winding (p, q) for each primitive normal (p, q) tothe toric diagram
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 17 / 49
Q: We have a NS5 cycle for each primitive normal of the toric diagram.Why?
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 18 / 49
Q: We have a NS5 cycle for each primitive normal of the toric diagram.Why?
The key is Buscher’s rule (’87,’88)
ds210 = ds29 + g99(dx9 + v1)2.
DecomposeB = b2 + b1 ∧ (dx9 + v1).
Then Buscher’s T-duality rule gives
v0
1 = b1, b0
1 = v1, b0
2 = b2 + b1 ∧ v1.
Roughly,
Buscher’s rule
metric ↔ B-field
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 18 / 49
By applying T-duality along 57-directions to toric Calabi-Yau cone, themetric becomes flat and instead we have the B-field
B¸ = v1 ∧ (dθ¸1 + v1) + v2 ∧ (dθ¸
2 + v1),
where v1 and v2 are gauge fields in 57-directions, respectively. This B-fielddepends on the facet α, and jumps as we go from one facet α to anotherα + 1:
B¸ → B¸+1 = B¸ − (∆p v1 + ∆q v2) ∧ dθ3,
This means the presence of NS5-brane at the intersection of NS5-branes,i.e. at the primitive normals of the toric diagram.
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 19 / 49
Zia-zag paths and fivebrane diagrams
Hanany-Vegh (’05) discovered a relation between ‘zig-zag paths’ of thebipartite graphs and the toric diagram.
In our explanation, this is not a conjecture and automatically follows fromour construction since ‘zig-zag paths’ is equivalent to cycles of NS5-branes.
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 20 / 49
So far as topological information of branes are concerned, we can usezig-zag paths, but the real brane configuration is different.
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 21 / 49
Notes on string coupling
Actually, our analysis so far is in the strong coupling limit.
Real shape of branes: difficult to determine in general string couplingconstant gs (we need to solve EOM), but can be analyzed in the limitof gs → 0 and gs → ∞:
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 22 / 49
Notes on string coupling
Actually, our analysis so far is in the strong coupling limit.
Real shape of branes: difficult to determine in general string couplingconstant gs (we need to solve EOM), but can be analyzed in the limitof gs → 0 and gs → ∞:
(Strong coupling limit) Take gs → ∞. Then
TD5 ≫ TNS5
and D5-branes become flat T2 and NS5-branes orthogonal to D5.
(Weak coupling limit) Take gs → 0. Then
TNS5 ≫ TD5
and N5-brane worldvolume Σ becomes a holomolophic curve
Σ : W(x, y) = 0 in(
Cˆ
)2,
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 22 / 49
Weak coupling limit
Consider the weak coupling limit gs → 0 (the decoupling limit). Then
TNS5 ≫ TD5
Then NS5-brane worldvolume Σ is a holomolophic curve W(x, y) = 0 in(
Cˆ
)2, where
x = exp(x4 + ix5), y = exp(x6 + ix7)
W(x, y) is a Newton Polynomial of the toric diagram
W(x, y) =∑
(i;j)2∆
c(i;j)xiyj
where ∆ ∈ Z2 is the toric diagram.
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 23 / 49
SU(N) gauge groups live on D5-brane discs Da’s, whose intersectionwith NS5-brane Σ gives one-cycles Ca on Σ.
Example: P1 × P
1
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 24 / 49
Reading off quiver gauge theories is completely analogous to the strongcoupling case.
D-branes Ca: gauge groups
intersection point of Ca: bifundamentals
discs surrounded by Ca’s: term in the superpotential
In fact, we can explain everything in the weak coupling limit.
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 25 / 49
Reading off quiver gauge theories is completely analogous to the strongcoupling case.
D-branes Ca: gauge groups
intersection point of Ca: bifundamentals
discs surrounded by Ca’s: term in the superpotential
In fact, we can explain everything in the weak coupling limit.
Problem: how to determine D5-brane cycles Ca from toric data?
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 25 / 49
Reading off quiver gauge theories is completely analogous to the strongcoupling case.
D-branes Ca: gauge groups
intersection point of Ca: bifundamentals
discs surrounded by Ca’s: term in the superpotential
In fact, we can explain everything in the weak coupling limit.
Problem: how to determine D5-brane cycles Ca from toric data?
Answer: topological date of Ca’s are given by the method of untwisting,relating weak coupling to strong coupling
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 25 / 49
Untwisting
Advantages and disadvantages
Strong coupling limit: direct relation with toric data
Weak coupling limit: gauge theory meaning clear
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 26 / 49
Untwisting
Advantages and disadvantages
Strong coupling limit: direct relation with toric data
Weak coupling limit: gauge theory meaning clear
There exists a method to relate weak coupling to strong coupling, which isknown as untwisting [Feng-He-Kennaway-Vafa ’05].
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 26 / 49
The zig-zag path, or the windingcycle of T
2 in bipartite graph, isturned into punctures of Σ in theweak coupling limit. Physically,this corresponds to semi-infinitecylinder of NS5-brane.
The face of bipartite graph on T2
in the strong coupling limit ismapped to a disc Da bounding1-cycle Ca of Σ. Physically, thiscorresponds to a stack of ND5-branes.
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 27 / 49
Example of untwisting: C3
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 28 / 49
Example of untwisting: F0 = P1 × P
1
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 29 / 49
Plan
1 Introduction
2 Foundations of brane tilings
3 Generalizations and applications of brane tilingsApplicationsGeneralizations
4 Summary
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 30 / 49
AdS/CFT
AdS/CFT correspondence: IIB on AdS5 × X5(X5: Sasaki-Einsteinmanifold) is dual to 4d N = 1 superconformal quiver gauge theories.[Morrirson-Plesser, Acharya-Fugyerioa-O’Farrill-Hull-Spence ’98]
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 31 / 49
AdS/CFT
AdS/CFT correspondence: IIB on AdS5 × X5(X5: Sasaki-Einsteinmanifold) is dual to 4d N = 1 superconformal quiver gauge theories.[Morrirson-Plesser, Acharya-Fugyerioa-O’Farrill-Hull-Spence ’98]
Brane tiling give the relation between gravity side (toricSasaki-Einstein manifold) and gauge theory (quiver) for infinitelymany examples.
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 31 / 49
Of course, this is NOT a derivation of AdS/CFT! We should try to verifythe proposal.Difficulty: superconformal field theory is strongly coupled
One possible check: use [Gubser ’98]
vol(X5) =π3
4
1
a
where a is the central charge of 4d superconformal quiver gauge theory.
‘proven’ for arbitary toric Calabi-Yau [Butti-Zaffaroni ’05]
volume: computed the volume minimization (localization)[Martelli-Sparks ’05, ’06]
central charge: computed by a-maximization [Intriligator-Wecht ’03]
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 32 / 49
Equivalence with mirror Calabi-Yau with intersectingD6-branes
Take T-duality along 8-directions. Then
D5-brane → D6-braneNS5-brane → mirror Calabi-Yau
here NS5-brane surface Σ : W(x, y) = 0 (x, y ∈ Cˆ) is replaced by
W(x, y) = uv,
with u, v ∈ C, which is the mirror Calabi-Yau [Hori-Iqbal-Vafa ’00]Thus we have intersecting D6-brane systems with mirror Calabi-Yau[Feng-He-Kennaway-Vafa ’05]
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 33 / 49
(Homological) mirror symmetry
(Homological) Mirror Symmetry: useful for studying Calabi-Yaumanifolds themselves. Completely rigorous mathematical proof isgiven from physics intuition [Ueda-M.Y. ’06, ’07]
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 34 / 49
(Homological) mirror symmetry
(Homological) Mirror Symmetry: useful for studying Calabi-Yaumanifolds themselves. Completely rigorous mathematical proof isgiven from physics intuition [Ueda-M.Y. ’06, ’07]
T−dual along 579-directions= Mirror Symmetry !
T−dual along 57-directions T−dual along 9-direction
D3−brane
probing CYD5/NS5
wrapping 3−cycles of CY
Intersecting D6−branes
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 34 / 49
Deformation of brane systems and marginal deformation ofN = 1 superconformal quiver gauge theories
The degrees of freedom of deformation of branes have a physicalmeaning as the marginal deformation of corresponding superconformalquiver gauge theory [Imamura-Isono-Kimura-M.Y. ’07]
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 35 / 49
Deformation of brane systems and marginal deformation ofN = 1 superconformal quiver gauge theories
The degrees of freedom of deformation of branes have a physicalmeaning as the marginal deformation of corresponding superconformalquiver gauge theory [Imamura-Isono-Kimura-M.Y. ’07]
This is probably the first discussion of real branes, not just graphs
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 35 / 49
More precisely,
diagonal gauge coupling ↔ C57 + ie`ffi
β-like deformation ↔ C + iB57
other d − 3 marginal deformations ↔ geometric deformation of branes
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 36 / 49
Fractional branes
We can change the rank of the gauge groups by including fractional branes.
Anomaly cancellation in gauge theory follows from charge conservation[Imamura ’06]
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 37 / 49
Flavor Branes
We can also consider flavor branes, which are D7-branes wrapping 4-cyclesof Calabi-Yau in D3-brane setup. By T-duality, they become D5-branesspaned between two NS5-branes. [Imamura-Kimura-M.Y. ’08]
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 38 / 49
Orientifold of brane tilings
We can also include orientifold planes [Franco et. al. ’07]. If you preserveSUSY then you can include O5-planes and O7-planes
0 1 2 3 4 5 6 7 8 9
D5 ◦ ◦ ◦ ◦ ◦ ◦NS5 ◦ ◦ ◦ ◦ ΣO5 ◦ ◦ ◦ ◦ ◦ ◦O7 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
Consider the case of O5-planes, which are represented as points on T2.
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 39 / 49
Gauge groups and matter representations are determined from ‘T-parities’(different from RR-charges) of O5-planes
S O (N )
+ on edge: symm. repr.
– on edge: anti-symm. repr.
+ on face: SO(N)
– on face: SP(N/2)
Many rules to determine T-parities were proposed by [Franco et. al. ’07],and derived and elaborated by [Imamura-Kimura-M.Y. ’08]
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 40 / 49
Charge conservation of branes and cancellation of gaugeanomalies
Orientifolded theories often have anomalies since
d = (N − 4)d , d = (N + 4)d .
and thus we have to include flavor branes.
We have shown [Imamura-Kimura-M.Y. ’08] that anomalycancellation in gauge theory is derived from the condition of chargeconservation in string theory
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 41 / 49
Models of Dynamical SUSY breaking
You can make models of dynamical SUSY breaking (e.g. (3,2)-model[Affleck-Din-Seiberg ]) from string theory [Franco et. al, ’07]!
You need to consider “flow of NS5-brane charge” [Imamura-Kimura-M.Y.’08]!
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 42 / 49
Summary
Brane tilings are physical brane systems consisting of D5-branes andNS5-branes. Part of their information is concisely summarized as bythe fivebrane diagram, or a bipartite graph on T
2.
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 43 / 49
Summary
Brane tilings are physical brane systems consisting of D5-branes andNS5-branes. Part of their information is concisely summarized as bythe fivebrane diagram, or a bipartite graph on T
2.
On the D5-brane worldvolume, we have 4d N = 1 superconformalquiver gauge theory.
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 43 / 49
Summary
Brane tilings are physical brane systems consisting of D5-branes andNS5-branes. Part of their information is concisely summarized as bythe fivebrane diagram, or a bipartite graph on T
2.
On the D5-brane worldvolume, we have 4d N = 1 superconformalquiver gauge theory.
By T-duality, brane tilings are equivalent to toric Calabi-Yau conesprobed by D3-branes.
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 43 / 49
Summary
Brane tilings are physical brane systems consisting of D5-branes andNS5-branes. Part of their information is concisely summarized as bythe fivebrane diagram, or a bipartite graph on T
2.
On the D5-brane worldvolume, we have 4d N = 1 superconformalquiver gauge theory.
By T-duality, brane tilings are equivalent to toric Calabi-Yau conesprobed by D3-branes.
Brane tilings are powerful because they apply to arbitrary toricCalabi-Yaus, and thus to a large class of quiver gauge theories.
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 43 / 49
Generalizations:
Fractional branes
Flavor branes
Orientifold . . .
Applications:
AdS/CFT correspondence
mirror symmetry . . .
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 44 / 49
Outlook
Real meaning of dimers (they are exactly solvable in 2d!)? Partionfunction of dimer model, perfect matchings?
Dynamics of N = 1 SYM, such as gaugino condensation, χSB,confinement? a-maximization?
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 45 / 49
Outlook: relation with many topics
Model building [Wijnholt, Uranga and many other people,...; alsoBerenstein-Pinansky ’06]: it is possible to realize models almostequivalent to standard models and GUT. Yukawa couplings fromD-brane instantonts? (although probably F-theory is better)
Mathematical applications mirror symmetry, Sasaki-Einstein geometry
Tropical geometry, amoeba: appears for example in vortex-instantonsystems in 5d N=1 SYM [Fujimori-Nitta-Ohta-Sakai-M.Y. ’08], stringjunctions [Lunin ’08]
Relation with dimer models in crystal melting (topological A-model)?Should be related [Feng-He-Kennaway-Vafa ’05], but how exactly?Relation with Donaldson-Thomas invariant [Szendroi ’07]
Generalization to non-toric case, such as non-toric del Pezzos?
Generalization to CY4 (AdS4/CFT3?) Relation with BLG theory? [S.Lee and collaborators ’06-’07, cf. recent paper by Hosomichi et. al.]
...
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 46 / 49
What can be learn from “brane tilings” in the future?
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 47 / 49
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 48 / 49
The strong coupling limit:
gstr → ∞, ls → 0, R → 0, withR2
gstrl2skept fixed . (1)
The weak coupling limit:
gstr → 0, ls → 0, R → 0, withR2
gstrl2skept fixed . (2)
Masahito Yamazaki (Univ. Tokyo) Brane Tilings 49 / 49