Gauge Theory and Topological Strings Geometry Conference in

Gauge Theory and Topological 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 Dijkgraaf University of Amsterdam

Nigel Hitchin’s Circle of Ideas

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

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

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

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

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

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

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

Topological String (A model) Quantum corrections, X t H 2(X, C) F gqu (t) = GWg; d e¡ dt d Partition function X ¸ 2 g¡ 2 F g (t) Z t op (t; ¸ ) = exp g F g (t) = F gcl (t) + F gqu (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. Mcomplex structure X moduli space

Mirror Symmetry A-model quantum B-Model classical

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

Mirror Symmetry B model A model Dual Torus Fibrations base

D-Branes B A homological mirror symmetry) coherent sheaves derived category special Lagrangians + gauge bundle Fukaya category

Charge Lattice (B-model) = H 3 (X ; Z) ¤ B = K 1 (X ) » Symplectic vector space = H 3 (X ; C ) V = ¤ C» Z ®^ ¯ X

Period Map & Quantization hol 3 -form dz 1 dz 2 dz 3 V moduli space of CY MX Lagrangian cone L=graph (d. F 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(2 n, Z) change of canonical basis (A, B)

complexified Kähler volume A-Model 1 k+ i B e ¸ F 0 cl symplectic vector space H ev (X ; C ) h®; ¯i = index D ® t 3 = 6¸ 2 + GW quantum corrections ¯¤ complexified. K ähler cone

Charged objects: D-branes electric-magnetic charges Large volume: • q electric D 0 -D 2 • p magnetic D 4 -D 6 charged particles

Gauge Theory Invariants Coherent sheaf Charge E! X ¹ = [E] 2 K 0 (X ) = ¤ (conjectured) Donaldson-Thomas invariant R D(¹ ) = 1 [M ¹ ]v i r Moduli space of stable sheaves

Gauge Theory D( ) is conjectured to be the partition function of a 6 -dim topologically twisted gauge theory Z S= j. F j 2 + j. D Ái j 2 + [Ái ; Áj ]2 + : : : X 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 gau ge (p; Á) = X q 2 ¤ ¡ D (p; q)ei q¢Á

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

Donaldson-Thomas Invariants U(1) gauge theory + singularities q=(d, n) n = ch 3 » Tr F 3 d = ch 2 » Tr F 2 instanton strings Z gau ge = X d; n D (d; n)edt + n ¸

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

Gopakumar-Vafa invariants M-theory limit ¸ ¸ ! 1 virtual loops of M 2 branes

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

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

Simplest Calabi-Yau

constant maps g

Stat-Mech: 3 d Partitions GW GV DT

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

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

Gauge theory D-brane Black hole

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

Microscopic counting (quantum) OSV S(p; q) = mi n Á f Im F t op (p + i Á) + qÁg

Attractor Mechanism near-horizon moduli

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

B-model

Hitchin’s theory of 3 -forms

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

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

Covers of T 2, repr of Sd Σg action of Z 2 twist field gas of branch points

Mirror symmetry: modular forms q = e 2¼i t

N D 4 -branes (m=1) U(N) Gauge Theory

Two-dimensional U(N) Yang-Mills

Wilson loop

N free non-relativistic fermions Z gau ge = X e¡ ¸ E + i µP f er m i on s C 2 = E = X i 1 2 pi ; 2 C 1 = P = X i pi

Ground state energy E 0 = I m(F cl ) F cl 3 1 cl t t = + F 0 (t) + F 1 cl (t) = ¡ ¸ 2 6¸ 2 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 dx x Y p 2 Z ¸ = exp X 0 + ³ 1 + xept + ´ ³ 1 2 ¸ p 2 1 2 ¸ 2 g¡ 2 F g (t) g quasi-mdoular form of wt 6 g-6 1 + x ¡ 1 ept ¡ ´ 1 2 ¸ p 2

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

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

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

Elliptic Genus Fourier expansion X D (d; n)e 2¼i dz e 2¼i n ¿ elll(M ; z; ¿) = d; n Modular properties Z gau ge (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!