Z THEORYZ THEORY

Nikita NekrasovNikita Nekrasov

IHES/ITEPIHES/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

Z THEORYZ THEORY

Interplay between Interplay between

(non-perturbative) topological strings (non-perturbative) topological strings

and and

topological gauge theorytopological gauge theory

Other names:

mathematicalmathematical M-theory,

topologicaltopological M-theory,

/-theory

TOPOLOGICAL STRINGSTOPOLOGICAL 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,…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 LagrangianIn string theory need something else:

Known sources of nonpert effectsD-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,

whose homology classes belong to

a Lagrangian sublattice in the middle dim homology

Subtleties in integration over the moduli of Lagrangain submanifolds. In the simplest cases reduces to the study of

Chern-Simons gauge theory on L

{L}

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

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

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

Replace Kodaira-Spencer Lagrangian which describes deformations of ( X,

BY LAGRANGIAN FOR

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 CASESum 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-DUALITYOPEN + CLOSED TYPE A

STRING ON X=

OPEN + CLOSED TYPE B STRING ON X

GW – DT correspondence

Choice of the lattice in the K(X):

Ch()

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 FOAMThe 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)

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

DW – GW correspondenceGauge group G corresponds to GW theory on

GEOMETRIC ENGINEERING OF 4d GAUGE THEORIESGEOMETRIC 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?

• 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;? ? ?? ?

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