Gauge Theory andTopological Strings

Geometry Conference in honour of Nigel Hitchin

- RHD, C. Vafa, E.Verlinde, hep-th/0602087 - J. de Boer, M. Chang, RHD, J. Manschot, E. Verlinde,

hep-th/0608059

Robbert DijkgraafUniversity of Amsterdam

Nigel Hitchin’s Circle of

Ideas

integrable systemsintegrable systemsspecial holonomiesspecial holonomies

hyper-Kahlerhyper-Kahler

calibrationscalibrations

(generalized) CY(generalized) CYspectral curvesspectral curves

mirror symmetrymirror symmetry

self-dual geometryself-dual geometry

quantization quantization

instantons instantons

Nigel Hitchin’s Circle of

Ideas

monopolesmonopoles

Higgs-bundlesHiggs-bundles

integrable systemsintegrable systemsspecial holonomiesspecial holonomies

hyper-Kahlerhyper-Kahler

calibrationscalibrations

(generalized) CY(generalized) CYspectral curvesspectral curves

mirror symmetrymirror symmetry

self-dual geometryself-dual geometry

quantization quantization

instantons instantonsmonopolesmonopoles

Higgs-bundlesHiggs-bundles

integrable systemsintegrable systemsspecial holonomiesspecial holonomies

hyper-Kahlerhyper-Kahler

calibrationscalibrations

(generalized) CY(generalized) CYspectral curvespectral curve

mirror symmetrymirror symmetry

self-dual geometryself-dual geometry

quantization quantization

instantons instantonsRandom Walkmonopolesmonopoles

Higgs-bundlesHiggs-bundles

X simply-connected Kähler manifold, dimC X=3, c1(X) = 0, no torsion.

X

Calabi-Yau threefolds

Diffeomorphism type of X is completely fixed by b3(X) and b2(X) plus classical invariants

F cl0 (t) =

Z

X

16t3; t 2 H 2(X ;Z)

F cl1 (t) =

112

Z

Xt ^c2

Decomposition

=

X 0X § g

X = X 0#§ g

b3 = 0 b2 = 0

Core

§ g = #g¡S3 £ S3

¢

[C.T.C. Wall]

Miles Reid’s Fantasy:“There is only one CY space”

M g

b2 = 0

All CY connected through conifoldtransitions S3 → S2

b2 = 1Kähler CYs

complexstructuremoduli

Gromov-Witten InvariantsExact instanton sum 2 ( , )d H X

genus g X

Moduli stack of stable maps

GWg;d =

Z

[M g (X ;d)]v i r

1 2 Q

Topological String (A model)

F qug (t) =

X

d

GWg;de¡ dt

Quantum corrections, tH2(X,C)

Ztop(t;¸) = expX

g

¸2g¡ 2Fg(t)

Partition function

Fg(t) = F clg (t) +F qu

g (t)

Topological String (B model)Complex moduli, tH2,1(X)

Localizes on (almost) constant maps df=0

f

Kodaira-Spencer field theory

• genus 0: classical Variation of Hodge Structures

• genus 1: analytic Ray-Singer torsion

• genus 2 and higher: quantum corrections

quantization of complex structuremoduli space M X

Mirror Symmetry

X

X

0 ( )F t

classical

/0 0,

0

( ) tdd

d

F t GW e

quantum

A-model B-Model

CY

3S

3T

CY fibered by special Lagrangian T3

[Strominger, Yau, Zaslov]

network ofsingularities

S1 shrinks

Mirror Symmetry

X X

Dual Torus Fibrations

2,1h

3T

X

base

1,1h

3 *( )T

X

base

A model B model

D-Branes

X0 ( ) ( )evenY K X H X

coherent sheaves

A

X

13( ) ( )Y K X H X

special Lagrangians+ gauge bundle

B

homological mirror

symmetry)

derived category Fukaya category

Symplectic vector space V = ¤ C »= H 3(X ;C)

Z

X®^¯

Charge Lattice (B-model)¤B = K 1(X ) »= H 3(X ;Z)

Period Map & Quantization

moduli space of CY MX

hol 3-form dz1 dz2 dz3

V

Lagrangian cone L=graph (dF0) semi-classical state ψ ~ exp F0

L

Special GeometryDarboux coordinates

3, ( , )jiA B H X

i

j

i

A

j B

q

p

0i i

Fp

q

3 CY X

Topological String Partition Function

2 2exp ( ), gtop g

g

Z F t

Transforms as a wave function (metaplectic representation) under Sp(2n,Z)

change of canonical basis (A,B)

A-Model

complexifiedKähler cone

symplectic vector space

complexifiedKähler volume

H ev(X ;C)

h®;¯i = index D® ¯ ¤

1¸

ek+iB

+ GW quantum corrections

F cl0 =

t3

6 2

Charged objects: D-branes

3CY time

charged particles

( , ) ( , )evp q H X

electric-magnetic charges

Large volume:•q electric D0-D2•p magnetic D4-D6

Gauge Theory Invariants

Coherent sheaf E ! X

(conjectured) Donaldson-Thomas invariant

D(¹ ) =R

[M ¹ ]v i r 1

Moduli space of stable sheaves

Charge ¹ = [E] 2 K 0(X ) = ¤

Gauge Theory

D() is conjectured to be the partition function of a 6-dim topologically twisted gauge theory

S =

Z

XjF j2 + jDÁi j2 + [Ái ;Áj ]2 + :::

Localizes to 6-dim version of Hitchin’s equations

Generating function

Choose polarization

K 0(X ) = ¤ = ¤+ ©¤¡

¹ = (p;q) 2 ¤

To make contact with GW-theory

¤+ = H 0 ©H 2; ¤¡ = H 4 ©H 6

Partition function

Zgauge(p;Á) =X

q2¤ ¡

D(p;q)eiq¢Á

GW-DT Equivalence[Maulik, Nekrasov, Okounkov, Pandharipande]

Consider the case of rank one, p = (1,0)(ideal sheaves)

where

Zgauge(p;Á) = Ztop(t;¸)

Á = (t;¸) 2 H 2(X ) ©H 0(X )

Donaldson-Thomas Invariants

U(1) gauge theory + singularities q=(d,n)

n = ch3 » TrF 3

d = ch2 » TrF 2

instanton strings

Zgauge =X

d;n

D(d;n)edt+n¸

Strong-weak coupling

GW ¸ ! 0

Ztop = expX

g;d

GWg;d¸2g¡ 2edt

DT ¸ ! 1

Zgauge =X

n;d

Dn;den¸ edt

Two expansion of single analytic function?

Gopakumar-Vafa invariants

charges q

( , ) log dettop qF t

1CY S

M-theory limit

virtual loopsof M2 branes

¸ ! 1

¸

GV Partition function

Gas of 5d charged & spinning black holes

Z(¸;t) =Y

n1;n2d ;m

³1¡ e (n1+n2+m)+td

´ ¡ N md

GV-invariants (integers)N m

d »p

d3 ¡ m2

Infinite products of Borcherds type.Automorphic properties?

Topological String Triality

top. stringsGromov-Witten

M-theoryGopakumar-

Vafa

gauge theoryDonaldson-

Thomas

3Simplest Calabi-Yau

31 2 3, ,z z z

2 2 23,0 1

2 (2 2)(2 2)!( )

g

g gg g

M g g g

B BGW c H

g

constant maps3

1 2

2

2,0

0

1( )

, 0

3d partitions

exp

1

g gtop g

g

n n

n n

Z GW

e

e

Stat-Mech: 3d Partitions

GW

GV

DT

Melting Crystals

Okounkov, Reshetikhin,Vafa, Nekrasov,...

OSV Conjecture[Ooguri, Strominger, Vafa]

Consider the limit

where

p! 1

Zgauge(p;Á) » jZtop(t;¸)j2

p+ iÁ =

µt¸

;1¸

¶2 H 2(X ) ©H 0(X )

3CY

3Black hole

Gauge theory

D-brane

Black Hole Entropy(semi-classical)

[Bekenstein, Hawking]

Microscopic counting (quantum)

S(p;q) = minÁ f ImFtop(p+ iÁ) + qÁg

OSV

charges ( , )p q

Attractor Mechanism

near-horizonmoduli

4d Black Holes

3CY time

Gauge Theory Top. Strings

large charges

Attractor CY

B-model

3( , ) ( , )P Q H X

, *X

S

3( , )H X

Hitchin’s theory of 3-forms

0

1 Im( )

2

Legendre

XS F

i

3 61 2 3( ) , e e e

( )i

0

d d

integrable complex structure

Integral structure

3( , )H X

“attractive”CY’s

3( , )H X

Bohr-Sommerfeld quantization of moduli space MX

Example of OSV conjectureLocal 2-torus in CY [Vafa]

area of T2 = t

2T X

Covers of T2, repr of Sd

2TΣg

Trdt

d

Z e eaction of Z2 twist field

gas of branch points

Mirror symmetry: modular forms

area t modulus t

q= e2¼it

N D4-branes

2T2

U(N) Gauge Theory

(m=1)

Two-dimensional U(N) Yang-Mills

2

2

1 F

T

S Tr F i

2 1( ) ( )

C R i C Rgauge

irrep R

Z e

1

12

0 0

0 0

0

exp

exp

ex0 p N

i

i

i

g U U

Wilson loop

P exp ( )g A U N

1i S

N free non-relativistic fermions

p

Zgauge =X

f er mions

e¡ ¸ E +iµP

C2 = E =X

i

12p2

i ; C1 = P =X

i

pi

Ground state energy

E0 = I m(F cl)

F cl =1¸2

F cl0 (t) + F cl

1 (t) = ¡t3

6 2+

t24

Black Hole statesYM instantons

/( ) ngauge

n

Z D n e

( )( )n

S nD n e

Free fermion states

Egauge

E

Z e

31

24N

gaugeZ e

Hardy-Ramanuyan, Rademacher, Cardy

Compute d(n) for large n?

Modular invariance

Grand canonical partition function

2 / 2 ( , , ) 1 N p ps N

N p

Z x g x Z x y q

, , ix e y e q e

ground state

excited state

p

N+

N

Large N limit: chiral factorization

2

gauge topZ Z

topZ topZ

Ztop(t;¸) =I

dxx

Y

p2Z ¸ 0+ 12

³1+ xept+ 1

2 ¸ p2´ ³

1+ x¡ 1ept¡ 12 ¸ p2

´

= expX

g

¸2g¡ 2Fg(t)

quasi-mdoular form of wt 6g-6

Two conjectures,related by modular transformation?

¸ ! 1=

OSV

Z(p;Á) » jZtop(t;¸)j2

p! 1

p+ iÁ =

µt¸

;1¸

¶

DT

Z(p;Á) = Ztop(t;¸)

p= 1

Á = (t;¸)

Rank zero, divisor

CY X

P

p= (0;c1) = (0;[P ]) q= (ch2;ch3) = (d;n)

Zgauge elliptic genus of modulus space M P

Elliptic GenusIf M is a CY k-fold

S1-equivariant y-genus of the loop space LM

weak Jacobi-form of wt 0 and index k/2

elll(M ;z;¿)

ell(¿;z + m¿ + n) = e¡ ¼id(m2¿+2mz)=2ell(¿;z)

ellµ

a¿ + bc¿ + d

;z

c¿ + d

¶

= e¼i k cz 22(c¿ + d) ell(¿;z)

Elliptic Genus

elll(M ;z;¿) =X

d;n

D(d;n)e2¼idze2¼in¿

Fourier expansion

Modular properties

Zgauge(t;¸) = ell(M P ;t;¸)

Topological String Theory

• Universal, deep, but mysterious object that captures many interesting connections between physics and geometry.

Happy Birthday, Nigel!