Gauge Theory and Topological Strings Geometry Conference in honour of Nigel Hitchin - RHD, C. Vafa, E.Verlinde, hep-th/ J. de Boer, M. Chang, RHD, J. Manschot, E. Verlinde, hep-th/ Robbert Dijkgraaf University of Amsterdam

Nigel Hitchin’s Circle of Ideas

integrable systems special holonomies hyper-Kahler calibrations (generalized) CY spectral curves mirror symmetry self-dual geometry quantization instantons Nigel Hitchin’s Circle of Ideas monopoles Higgs-bundles

integrable systems special holonomies hyper-Kahler calibrations (generalized) CY spectral curves mirror symmetry self-dual geometry quantization instantons monopoles Higgs-bundles

integrable systems special holonomies hyper-Kahler calibrations (generalized) CY spectral curve mirror symmetry self-dual geometry quantization instantons Random Walk monopoles Higgs-bundles

X simply-connected Kähler manifold, dim C X=3, c 1 (X) = 0, no torsion. X Calabi-Yau threefolds Diffeomorphism type of X is completely fixed by b 3 (X) and b 2 (X) plus classical invariants F c l 0 ( t ) = Z X 1 6 t 3 ; t 2 H 2 ( X ; Z ) F c l 1 ( t ) = 1 12 Z X t ^ c 2

Decomposition = X 0 X § g X = X 0 # § g b 3 = 0 b 2 = 0 Core § g = # g ¡ S 3 £ S 3 ¢ [C.T.C. Wall]

Miles Reid’s Fantasy: “ There is only one CY space ” M g b 2 = 0 All CY connected through conifold transitions S 3 → S 2 b 2 = 1 Kähler CYs complex structure moduli

Gromov-Witten Invariants Exact instanton sum X Moduli stack of stable maps GW g ; d = Z [ M g ( X ; d )] v i r 1 2 Q

Topological String (A model) F qu g ( t ) = X d GW g ; d e ¡ d t Quantum corrections, t  H 2 (X,C) Z t op ( t ; ¸ ) = exp X g ¸ 2 g ¡ 2 F g ( t ) Partition function F g ( t ) = F c l g ( t ) + F qu g ( t )

Topological String (B model) Complex moduli, t  H 2,1 (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 structure moduli space M X

Mirror Symmetry classical quantum A-model B-Model

CY fibered by special Lagrangian T 3 [Strominger, Yau, Zaslov] network of singularities S 1 shrinks

Mirror Symmetry Dual Torus Fibrations base A model B model

D-Branes coherent sheaves A 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 M X hol 3-form dz 1  dz 2  dz 3 V Lagrangian cone L=graph (dF 0 )  semi-classical state ψ ~ exp F 0

Special Geometry Darboux coordinates

Topological String Partition Function Transforms as a wave function (metaplectic representation) under Sp(2n,Z) change of canonical basis (A,B)

A-Model complexified Kähler cone symplectic vector space complexified Kähler volume H ev ( X ; C ) h ® ; ¯ i = i n d ex D ® ­ ¯ ¤ 1 ¸ e k + i B + GW quantum corrections F c l 0 = t 3 6 ¸ 2

Charged objects: D-branes charged particles 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 X j F j 2 + j D Á i j 2 + [ Á i ; Á j ] 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 Z gauge ( p ; Á ) = X q 2 ¤ ¡ D ( p ; q ) e i q ¢ Á

GW-DT Equivalence [Maulik, Nekrasov, Okounkov, Pandharipande] Consider the case of rank one, p = (1,0) (ideal sheaves) where Z gauge ( p ; Á ) = Z t op ( t ; ¸ ) Á = ( t ; ¸ ) 2 H 2 ( X ) © H 0 ( X )

Donaldson-Thomas Invariants U(1) gauge theory + singularities q=(d,n) n = c h 3 » T r F 3 d = c h 2 » T r F 2 i ns t an t ons t r i ngs Z gauge = X d ; n D ( d ; n ) e d t + n ¸

Strong-weak coupling GW ¸ ! 0 Z t op = exp X g ; d GW g ; d ¸ 2 g ¡ 2 e d t DT ¸ ! 1 Z gauge = X n ; d D n ; d e n ¸ e d t Two expansion of single analytic function?

Gopakumar-Vafa invariants M-theory limit virtual loops of M2 branes ¸ ! 1 ¸

GV Partition function Gas of 5d charged & spinning black holes Z ( ¸ ; t ) = Y n 1 ; n 2 d ; m ³ 1 ¡ e ¸ ( n 1 + n 2 + m ) + t d ´ ¡ N m d GV-invariants (integers) N m d » p d 3 ¡ m 2 Infinite products of Borcherds type. Automorphic properties?

Topological String Triality top. strings Gromov-Witten M-theory Gopakumar-Vafa gauge theory Donaldson-Thomas

Simplest Calabi-Yau

g constant maps

Stat-Mech: 3d Partitions GW GV DT

Melting Crystals Okounkov, Reshetikhin, Vafa, Nekrasov,...

OSV Conjecture [Ooguri, Strominger, Vafa] Consider the limit where p ! 1 Z gauge ( p ; Á ) » j Z t op ( t ; ¸ )j 2 p + i Á = µ t ¸ ; 1 ¸ ¶ 2 H 2 ( X ) © H 0 ( X )

Black hole Gauge theory D-brane

Black Hole Entropy (semi-classical) [ Bekenstein, Hawking ]

Microscopic counting (quantum) S ( p ; q ) = m i n Á f I m F t op ( p + i Á ) + q Á g OSV

Attractor Mechanism near-horizon moduli

4d Black Holes Gauge Theory Top. Strings large charges Attractor CY

B-model

Hitchin’s theory of 3-forms

Integral structure “attractive” CY’s Bohr-Sommerfeld quantization of moduli space M X

Example of OSV conjecture Local 2-torus in CY [ Vafa ] area of T 2 = t X

Covers of T 2, repr of S d ΣgΣg action of Z 2 twist field gas of branch points

Mirror symmetry: modular forms q = e 2 ¼ i t

N D4-branes U(N) Gauge Theory (m=1)

Two-dimensional U(N) Yang-Mills

Wilson loop

N free non-relativistic fermions Z gauge = X f erm i ons e ¡ ¸ E + i µ P C 2 = E = X i 1 2 p 2 i ; C 1 = P = X i p i

Ground state energy E 0 = I m ( F c l ) F c l = 1 ¸ 2 F c l 0 ( t ) + F c l 1 ( t ) = ¡ t 3 6 ¸ 2 + t 24

Black Hole states YM instantons Free fermion states

Hardy-Ramanuyan, Rademacher, Cardy Compute d(n) for large n? Modular invariance

Grand canonical partition function

ground state excited state

Large N limit: chiral factorization

Z t op ( t ; ¸ ) = I d x x Y p 2 Z ¸ ³ 1 + xe p t ¸ p 2 ´³ 1 + x ¡ 1 e p t ¡ 1 2 ¸ p 2 ´ = exp X g ¸ 2 g ¡ 2 F g ( t ) quasi-mdoular form of wt 6g-6

Two conjectures, related by modular transformation? ¸ ! 1 = ¸ OSV Z ( p ; Á ) » j Z t op ( t ; ¸ )j 2 p ! 1 p + i Á = µ t ¸ ; 1 ¸ ¶ DT Z ( p ; Á ) = Z t op ( t ; ¸ ) p = 1 Á = ( t ; ¸ )

Rank zero, divisor CY X P p = ( 0 ; c 1 ) = ( 0 ; [ P ]) q = ( c h 2 ; c h 3 ) = ( d ; n ) Z gauge elliptic genus of modulus space M P

Elliptic Genus If M is a CY k-fold S 1 -equivariant  y -genus of the loop space LM weak Jacobi-form of wt 0 and index k/2 e lll ( M ;z ; ¿ ) e ll ( ¿ ; z + m¿ + n ) = e ¡ ¼ i d ( m 2 ¿ + 2 mz )= 2 e ll ( ¿ ; z ) e ll µ a¿ + b c¿ + d ; z c¿ + d ¶ = e ¼ i k cz 2 2 ( c¿ + d ) e ll ( ¿ ; z )

Elliptic Genus e lll ( M ;z ; ¿ ) = X d ; n D ( d ; n ) e 2 ¼ i d z e 2 ¼ i n¿ Fourier expansion Modular properties Z gauge ( t ; ¸ ) = e ll ( M P ; t ; ¸ )

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

Happy Birthday, Nigel!