Brane Tilings and Homological Mirror Symmetry · Use Brane Tilings to Prove (Homological) Mirror...

Brane Tilings andHomological Mirror Symmetry

Masahito Yamazaki

Univ. of Tokyo (Hongo)

Based on math.AG/0605780, 0606548, 07?????(In collaboration with Kazushi Ueda)

2006/12/06 @ Tokyo Institute of Technology – p.1/28

Introduction

Recent development: "Brane Tiling" (introduced in2005), a bipartite graph (dimer) on torus:


Brane Tiling

As I will explain, brane tilings represent configuration ofD5-branes and NS5-branes.

On the world-volume of these D5-branes, we haveN = 1 superconformal quiver gauge theory , a versionof gauge theory specified by an oriented graph(quiver ).

The brane tiling technique makes it possible toconstruct brane configurations for complicated quivergauge theories.


Various Realizations in String Theory

Actually, quiver gauge theories have severalrealizations in string theory, which are all related byT-duality, and brane tiling is only one of them. (D3,D4/NS5, D5/NS5, D6...)

Brane Tiling

D3−braneprobing CY

D5/NS5wrapping 3−cycles of CY

D6−brane

T−dualT−dual (*2)



Among these, D3-brane picture and D6-brane pictureare related by T-duality 3 times, which is nothing butmirror transformation.

T−dual (*3)=Mirror

D3−braneprobing CY wrapping 3−cycles of CY

D6−brane



This means D5/NS5-system (brane tiling) might beuseful for investigating mirror symmetry.

D3−braneprobing CY wrapping 3−cycles of CY

D6−braneD5/NS5

T−dual (*3)=Mirror

T−dual (*2) T−dual

Our conclusion is that this is indeed the case!!


Today’s Goal

bababababababababababababababUse Brane Tilings to Prove (Homological) MirrorSymmetry

We will show that this method gives

Intuitive understanding from D-brane perspective

Rigorous mathematical proofs

Generalization to orbifold case (new result)


Plan of This Talk

1. Introduction

2. D-brane Realizations of Quiver Gauge Theories

(a) D3-brane Probing CY(b) D5/NS5-picture (Brane Tiling)(c) D6-brane Wrapping 3-cycles of CY

3. Proof of Homological Mirror Symmetry

4. Summary and Outlook


What is Quiver Gauge Theory?

Quiver(箙): "portable case for holding arrows", an oriented graph

vertex= gauge group

oriented arrow=bifundamental

(N1, N2)

U(N1) U(N2)bifundamental

each quiver specifies gauge theory

C3/Z3 F0 = P1�P1


D3-brane Probing Singular Calabi-Yau

Consider non-compact CY which has cone singularity.

We further assume that CY is toric (specified by toricdiagram).

D3-brane is transverse to CY and placed at the apex ofCY cone.

Sasaki−Einstein

D3−branetransverse to singularCalabi−Yau

Tip of Calabi−Yau cone

Calabi−Yau Cone


D3-brane Probing Singular Calabi-Yau

Then it is long belived that we have 4dN = 1 quivergauge theory on D3-brane.

Q: Which CY gives which quiver (and superpotential) ?A: Given by brane tiling

Sasaki−Einstein

D3−branetransverse to singularCalabi−Yau

Tip of Calabi−Yau cone

Calabi−Yau Cone


D5/NS5-System

If we T-dualize along 2-cycle of CY, then we haveconfiguration of D5-branes and NS5-branes:

0 1 2 3 4 5 6 7 8 9

D5 Æ Æ Æ Æ Æ ÆNS5 Æ Æ Æ Æ S (2-dim surface)

D5-brane worldvolume: R4� T2

NS5-brane worldvolume: R4� SReal shape of branes: difficult to determine in general(we need to solve EOM), but can be analyzed whengs ! 0 and gs ! ¥.


Strong Coupling

Consider the strong coupling limit gs ! ¥. Then

TD5 � TNS5

Then D5-branes become flat and NS5-branes areorthogonal to D5.

Stack of N D5-branes are divided by NS5-branes intoseveral regions, thus we have multiple gauge groups.

NS5 D5

NS5


Strong Coupling

Due to conservation of NS-charge, D5-brane actuallybecomes (N, k)-branes. (k = 1, 0,�1 in this talk)

Dimer Model

White Region: (N,0)−brane

BlueRed

Region: (N,1)−brane Region: (N,−1)−brane

2

3 3

1

44

4 4

6

7

1

5 5


Strong Coupling

We have a bifundamental for each intersection pt of(N, 0)-branes.

From this we can read off quiver!

Quiver

White Region: (N,0)−brane

BlueRed

Region: (N,1)−brane Region: (N,−1)−brane

2

3 3

1

44

4 4

6

7

1

5 51

2 3


Weak Coupling

Consider the weak coupling limit gs ! 0. Then

TNS5 � TD5

Then NS5-brane worldvolume S is a holomolophic curve

W(x, y) = 0 in

�C��2, where

x = exp(x4 + ix5), y = exp(x6 + ix7)W(x, y) is a Newton Polynomial of the toric diagram

W(x, y) = å(i,j)2D c(i,j)xiyj

where D 2 Z2 is the toric diagram.


Untwisting

Actually, there exists a method to relate weak coupling tostrong coupling, which is known as untwisting[Feng-He-Kennaway-Vafa].


Example of untwisting: C3


Example of untwisting: C3


Example of untwisting: F0 = P1�P1


Example of untwisting: F0 = P1�P1


Result of Untwisting

The surface S we obtain after twisting is the curve ofNS5-brane (i.e. W(x, y) = 0). Therefore, byuntwisting, we can go to weak coupling.

Winding cycles of T2 are mapped to boundaries of SD5-branes (regions of T2) are mapped to windingcycles of S.


D6-brane Picture

Suppose we have gs ! 0 (weak coupling). Then, aswe have seen, NS5-brane is Riemann surfaceS : W(x, y) = 0.

Take T-duality, then NS5-brane turns into CY, which isoften written as double fibration over W-plane.

W = W(x, y), W = uv

where w, z 2 C� and u, v 2 C.

D5-brane is mapped to D6-branes wrapping 3-cyclesof CY.

Now we have D6-branes wrapping 3-cycles of CY!!


Plan of This Talk

1. Introduction



3. Proof of Homological Mirror Symmetry4. Summary and Outlook


Mirror Symmetry

Mirror Symmetry:

B-model on CY = A-model on another CY

How to give a mathematical formulation of mirrorsymmetry?


Homological Mirror Symmetry

Partial Answer: Homological Mirror Symmetry [Kontsevich]:

Db(coh Y)| {z }(B�brane) �= Db Fuk!(W)| {z }(A�brane) ,

Db(coh Y): derived category of coherent sheavesFuk!(W): (directed) Fukaya category

It is known that B-model side can be computed fromexceptional collections (Bondal’s theorem)

A-model side (Fukaya category) is much more difficultto compute


What is Category?

Fukaya category is a so-called A¥-category

(Roughly speaking) an A¥-category consists of1. Objects: Oi

2. Morphism: Mor(O1,O2) for 2 objects O1 and O2

3. Composition of Morphism:mk : Mor(O0,O1)�Mor(O1,O2)� . . .�Mor(Ok�1,Ok)! Mor(O0,Ok)


What is Fukaya Category?

1. Objects: vanishing cycles fCigNi=1

(Mirror fiber degenerates at critical points of W and atW = 0)

=VanishingCycle W=uv

W=W(x,y)

W=0 W=W* W=0 W=W*

3−cycle(Lagrangian)

D6-branes wrap these 3-cycles: Thus Objects=D-branes



2. Morphism: Mor(Ci, Cj) is the intersection points of Ci

and Cj. (open strings)3. Composition: non-zero when we can span a(pseudo-homorphic) disk. (disk amplitude)

Mor(C1, C3) = r

C1

C2

C3

Mor(C1, C2) = p

Mor(C2, C3) = qm2(p, q) = r

p

r

q



3. Composition: non-zero when we can span a(pseudo-homorphic) disk. (disk amplitude)

pk

pk+1

p4

mk(p1, p2, . . . pk) = pk+1

p1

p2

p3

m3(p, q, r) = s

u

p

s

t

r

q

but m3(u, p, s) = 0


Summary of Strategy

1. From toric diagram, draw D5/NS5 configuration instrong coupling (graph on T2).

2. Untwist to go to weak coupling. (graph on S)

3. Read off Fukaya Category.


Mathematical Formulation

This intuition can be given directly translated intomathematical formalism and now we have rigorousmathematical theorem!!

Main Theorem [Ueda-M.Y, math.AG/0606548]� �

Homological Mirror Symmetry is correct for P1�P1 andtheir Zn orbifold.� �


Bonus: Generalization to Zn Orbifold

Orbifolding by Zn simply corresponds to enlarging thefundemental domain by n.C3/Z3

Enlarge the Fundamental Domainby three times

C3

This means we can almost trivially extend our argument toZn orbifold case!


Plan of This Talk

1. Introduction



3. Proof of Homological Mirror Symmetry

4. Summary and Outlook


Summary

Brane tilings represent D5/NS5-brane system, andrelated by T-duality to Calabi-Yau geometry.

Brane tiling technology is useful to prove homologicalmirror symmetry

This method gives

Intuitive understanding from D-brane perspectiveRigorous mathematical proofsGeneralization to orbifold case (new result)


Outlook

More general toric CY (e.g. toric del Pezzo), (work inprogess with Kazushi Ueda)

Moduli spaces of brane tilings? "Phases" ofN = 1quiver gauge theories? (work in progress withY.Imamura, H.Isono, and K.Kimura)

Other dimensions? (cf. 3d brane tiling [Lee])

Dynamical SUSY breaking, metastable vacuum,gaugino condensation from brane tiling and D-branes?

More direct relation with BPS state counting as intopological vertex, instanton couting in SYM (cf.amoeba, tropical geometry?)

Relation with tachyon condensation?


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