Z THEORY Nikita Nekrasov IHES/ITEP Nagoya, 9 December 2004

Based on the work done in collaboration with: A.Losev, A.Marshakov, D.Maulik, A.Okounkov, H.Ooguri, R.Pandharipande, C.Vafa 2002-2004

(non-perturbative) topological strings topological gauge theory Z THEORY Interplay between (non-perturbative) topological strings and topological gauge theory Other names: mathematical M-theory, topological M-theory, m/f-theory

TOPOLOGICAL STRINGS Special amplitudes in Type II superstring compactifications on Calabi-Yau threefolds Simplified string theories, interesting on their own Mathematically better understood Come in several variants: A, B, (C…), open, closed,…

PERTURBATIVE vs NONPERTURBATIVE Usual string expansion: perturbation theory in the string coupling

NONPERTURBATIVE EFFECTS In field theory: from space-time Lagrangian In string theory need something else: Known sources of nonpert effects D-branes and NS-branes This lecture will not mention NS branes, except for fundamental strings

A model

A model

! D-branes in A-model Sum over Lagrangian submanifolds in X, Subtleties in integration over the moduli of Lagrangain submanifolds. In the simplest cases reduces to the study of Chern-Simons gauge theory on L Sum over Lagrangian submanifolds in X, whose homology classes belong to a Lagrangian sublattice in the middle dim homology

ALL GENUS A STRING ``Theory of Kahler gravity’’ Only a few terms in the large volume expansion are known For toric varieties one can write down a functional which will reproduce localization diagrams: could be a useful hint

B STORY Genus zero part = classical theory of variations of Hodge structure (for Calabi-Yau’s) Generalizations: Saito’s theory of primitive form, Oscillating integrals – singularity theory; noncommutative geometry;gerbes.

D-branes in B model Derived category of the category of coherent sheaves Main examples: holomorphic bundles ideal sheaves of curves and points D-brane charge: the element of K(X). Chern character in cohomology H*(X)

All genus B closed string KODAIRA-SPENCER THEORY OF GRAVITY CUBIC FIELD THEORY (+) NON-LOCAL (mildly +/- ) BACKGROUND DEPENDENT (-) NO IDEA ABOUT THE NON-PERTURBATIVE COMPLETION

B open string field theory HOLOMORPHIC CHERN-SIMONS W = holomorphic (3,0) – form on the Calabi-Yau X THIS ACTION IS NEVER GAUGE INVARIANT: NEED TO COUPLE B TO B*

B open string field theory CHERN-SIMONS F = closed 3– form on the Calabi-Yau X THIS ACTION IS GAUGE INVARIANT GENERALIZE TO SUPERCONNECTIONS TO GET THE OBJECTS IN DERIVED CATEGORY

HITCHIN’S GRAVITY IN 6d BY LAGRANGIAN FOR F Replace Kodaira-Spencer Lagrangian which describes deformations of ( X, W ) BY LAGRANGIAN FOR F

HITCHIN’S GRAVITY IN 6d

HITCHIN’S GRAVITY IN 6d

NAÏVE EXPECTATION Full string partition function = Perturbative disconnected partition function X D-brane partition function Z (full) = Z(closed) X Z (open) ???

D-brane partition function Sum over (all?) D-brane charges Integrate (what?) over the moduli space (?) of D-branes with these charges ? ? ? ? ? ? ? ?

Particular case of B-model D-brane counting problem Donaldson-Thomas theory Counting ideal sheaves: torsion free sheaves of rank one with trivial determinant

LOCALIZATION IN THE TORIC CASE Sum over torus-invariant ideals: melting crystals

Monomial ideals in 2d

DUALITIES IN TOPOLOGICAL STRINGS T-duality (mirror symmetry) S-duality INSPIRED BY THE PHYSICAL SUPERSTRING DUALITIES HINTS FOR THE EXISTENCE OF HIGHER DIMENSIONAL THEORY

T-DUALITY CLOSED/OPEN TYPE A TOPOLOGICAL STRING ON = CLOSED/OPEN TYPE B TOPOLOGICAL STRING ON

T-DUALITY Complex structure moduli of = Complexified Kahler moduli of AND VICE VERSA

S-DUALITY OPEN + CLOSED TYPE A STRING ON X = OPEN + CLOSED TYPE B

Choice of the lattice L in the K(X): GW – DT correspondence Choice of the lattice L in the K(X): Ch(L)

Describing curves using their equations ENUMERATIVE PROBLEM Virtual fundamental cycle in the Hilbert scheme of curves and points For CY threefold: expected dim = 0 Generating function of integrals of 1

GW – DT correspondence: degree zero Partition function = sum over finite codimension monomial ideals in C[x,y,z] = sum over 3d partitions = a power of MacMahon function: COINCIDES WITH DT EXPRESSION AS AN ASYMPTOTIC SERIES

QUANTUM FOAM The Donaldson-Thomas partition function can be interpreted as the partition function of Kahler gravity theory; Important lesson: metric only exists in the asymptotic expansion in string coupling constant. In the DT expansion: ideal sheaves (gauge theory)

K-THEORETIC GENERALIZATION ON TO SEVEN DIMENSIONS DT THEORY HAS A NATURAL K-THEORETIC GENERALIZATION CORRESPONDS TO THE GAUGE THEORY ON

DONALDSON-WITTEN FOUR DIMENSIONAL GAUGE THEORY Gauge group G (A, B, C, D, E, F, G - type) Z - INSTANTON PARTITION FUNCTION GEOMETRY EMERGING FROM GAUGE THEORY (DIFFERENT FROM AdS/CFT): Seiberg-Witten curves, as limit shapes

GEOMETRIC ENGINEERING OF 4d GAUGE THEORIES DW – GW correspondence Gauge group G corresponds to GW theory on GEOMETRIC ENGINEERING OF 4d GAUGE THEORIES

INSTANTON partition function

DW – GW correspondence

DW– GW correspondence In the G = SU(N) case the instanton partition function can be evaluated explicitly (random 2d partitions) Admits a generalization (higher Casimirs – Chern classes of the universal bundle) The generalization is non-trivial for N=1 (Hilbert scheme of points on the plane) Maps to GW theory of the projective line

RANDOM PARTITIONS Fixed point formula for Z, for G=SU(N): The sum over N-tuples of partitions The sum has a saddle point: limit shape It gives a geometric object: Seiberg-Witten curve: the mirror to

WHAT IS Z THEORY? TO BE CONTINUED......... Dualities + Unification of t and s moduli (complex and Kahler) suggest a theory of closed 3-form in 7-dimensions, or some chiral theory in 8d Candidates on the market: 3-form Chern-Simons in 7d coupled to topological gauge theory; Hitchin’s theory of gravity in 7d coupled to the theory of associative cycles; ? ? ?? ? TO BE CONTINUED.........

MAKE THEM SPECIAL HOLONOMY SPACETIMES….. FOR BETTER TIMES….. MAKE THEM SPECIAL HOLONOMY SPACETIMES…..