INSTANTON PARTITION FUNCTIONS Nikita Nekrasov IHES (Bures-sur-Yvette) & ITEP (Moscow)QUARKS-2008 May 25, 2008 Nikita Nekrasov IHES (Bures-sur-Yvette) & ITEP (Moscow)QUARKS-2008 May 25, 2008

Biased list of refs NN, NN, A.Aleksandrov~2008; NN, A.Marshakov~2006; A.Iqbal, NN, A.Okounkov, C.Vafa~2004; A.Braverman ~2004; NN, A.Okounkov ~2003; H.Nakajima, K.Yoshioka ~2003; A.Losev, NN, A.Marshakov ~2002; NN, 2002; A.Schwarz, NN, 1998; G.Moore, NN, S.Shatashvili ~ ; A.Losev, NN, S.Shatashvili ~ ; A.Gerasimov, S.Shatashvili ~ NN, NN, A.Aleksandrov~2008; NN, A.Marshakov~2006; A.Iqbal, NN, A.Okounkov, C.Vafa~2004; A.Braverman ~2004; NN, A.Okounkov ~2003; H.Nakajima, K.Yoshioka ~2003; A.Losev, NN, A.Marshakov ~2002; NN, 2002; A.Schwarz, NN, 1998; G.Moore, NN, S.Shatashvili ~ ; A.Losev, NN, S.Shatashvili ~ ; A.Gerasimov, S.Shatashvili ~

Mathematical problem: counting Integers: 1,2,3,….

Mathematical problem: counting Integers: 1,2,3,….

Mathematical problem: counting Partitions of integers: (1) (2) (1,1) (3) (2,1) (1,1,1) … Partitions of integers: (1) (2) (1,1) (3) (2,1) (1,1,1) …

Mathematical problem: counting Partitions of integers: (1) (2) (1,1) (3) (2,1) (1,1,1) … Partitions of integers: (1) (2) (1,1) (3) (2,1) (1,1,1) …

Mathematical problem: generating functions

Euler

Unexpected symmetry Dedekind eta

More structure: Arms, legs, and hooks

Growth process

Plancherel measure

Mathematical problem: counting Plane partitions of integers: ((1)); ((2)),((1,1)),((1),1); ((3)),((2,1)),((1,1,1)),((2),(1)),((1),(1),(1));…. Plane partitions of integers: ((1)); ((2)),((1,1)),((1),1); ((3)),((2,1)),((1,1,1)),((2),(1)),((1),(1),(1));….

Mathematical problem: counting Plane partitions of integers: ((1)); ((2)),((1,1)),((1),1); ((3)),((2,1)),((1,1,1)),((2),(1)),((1),(1),(1));…. Plane partitions of integers: ((1)); ((2)),((1,1)),((1),1); ((3)),((2,1)),((1,1,1)),((2),(1)),((1),(1),(1));….

Mathematical problem: generating functions MacMahon

Mathematical problem: more structural counting

Quantum gauge theory Four dimensions

Quantum gauge theory Four dimensions

Quantum sigma model Two dimensions

Quantum sigma model Two dimensions

Instantons Minimize Euclidean action in a given topology of the field configurations Gauge instantons (Almost) Kahler target sigma model instantons

Counting Instantons Approximation for ordinary theories. Sometimes exact results for supersymmetric theories.

Counting Instantons Approximation for ordinary theories. Sometimes exact results for supersymmetric theories.

Instanton partition functions in four dimensions Supersymmetric N=4 theory (Vafa-Witten)

Instanton partition functions in four dimensions Supersymmetric N=4 theory (Vafa-Witten) Transforms nicely under a (subgroup of) SL(2, Z)

Instanton partition functions in four dimensions Supersymmetric N=4 theory (Vafa-Witten) Transforms nicely under a (subgroup of) SL(2, Z) Hidden elliptic curve:

Instanton partition functions in four dimensions Supersymmetric N=2 theory (Donaldson-Witten) Supersymmetric N=2 theory (Donaldson-Witten) Intersection theory on the moduli space of gauge instantons

Instanton partition functions in four dimensions Supersymmetric N=2 theory (Donaldson-Witten) Supersymmetric N=2 theory (Donaldson-Witten) Donaldson invariants of four-manifolds Seiberg-Witten invariants of four-manifolds

Instanton partition functions in four dimensions Supersymmetric N=2 theory On Euclidean space R 4 Supersymmetric N=2 theory On Euclidean space R 4

Instanton partition functions in four dimensions Supersymmetric N=2 theory On Euclidean space R 4 Boundary conditions at infinity SO(4) Equivariant theory Supersymmetric N=2 theory On Euclidean space R 4 Boundary conditions at infinity SO(4) Equivariant theory

Instanton partition function Supersymmetric N=2 theory on Euclidean space R 4

Instanton partition function Supersymmetric pure N=2 super YM theory on Euclidean space R 4 Degree = Element of the ring of fractions of H*(BH) H = G X SO(4), G - the gauge group

Instanton partition function Supersymmetric N=2 super YM theory with matter

Instanton partition function Supersymmetric N=2 super YM theory with matter

Instanton partition function Supersymmetric N=2 super YM theory with matter Bundle of Dirac Zero modes In the instanton background

Instanton partition function Explicit evaluation using localization For pure super Yang-Mills theory:

Instanton partition function Compactification of the instanton moduli space to Add point-like instantons + extra stuff

Instanton partition function

For G = U(N)

Instanton partition function Perturbative part (contribution of a trivial connection) For G = U(N)

Instanton partition function Instanton part For G = U(N) Sum over N-tuples of partitions

Instanton partition function Generalized growth model

Instanton partition function Generalized growth model

Instanton partition function Generalized growth model

Instanton partition function Generalized growth model

Instanton partition function Generalized growth model

Instanton partition function Generalized growth model

Instanton partition function Limit shape Emerging geometry

Instanton partition function Limit shape Emerging algebraic geometry

Instanton partition function Limit shape Emerging algebraic geometry NN+A.Okounkov

Instanton partition function Limit shape Seiberg-Witten geometry NN+A.Okounkov

Instanton partition function Limit shape Seiberg-Witten geometry Integrability: Toda chain, Calogero-Moser particles, spin chains Hitchin system

Instanton partition function The full instanton sum has a hidden infinite dimensional symmetry algebra The full instanton sum has a hidden infinite dimensional symmetry algebra

Instanton partition function Special rotation parameters SU(2) reduction Special rotation parameters SU(2) reduction

Instanton partition function Fourier transform (electric-magnetic duality) Fourier transform (electric-magnetic duality)

Instanton partition function Fourier transform (electric-magnetic duality) Fourier transform (electric-magnetic duality)

Instanton partition function Free fermion representation J(z) form level 1 affine su(N) current algebra

Instanton partition function Free fermion representation

Instanton partition function Theory with matter in adjoint representaton Theory with matter in adjoint representaton That elliptic curve again

Instanton partition function Abelian theory with matter in adjoint representaton: back to hooks Abelian theory with matter in adjoint representaton: back to hooks

Instanton partition function Amazingly this partition function is also almost modular Amazingly this partition function is also almost modular

Instanton partition function Full-fledged partition function: Generic rotations and fifth dimension K-theoretic version Full-fledged partition function: Generic rotations and fifth dimension K-theoretic version

Instanton partition function Free field representation: Infinite product formula Free field representation: Infinite product formula

Instanton partition function Free fields and modularity: Infinite product of theta functions Free fields and modularity: Infinite product of theta functions

Instanton partition function Free field representation: Second quantization representation Free field representation: Second quantization representation

Instanton partition function Free field representation: Second quantization representation Free field representation: Second quantization representation Bosons (+) and fermions (-)

Instanton partition function Free fields? Where? What kind?

M-theory to the rescue The kind of instanton counting we encountered occurs naturally in the theory of D4 branes in IIA string theory to which D0 branes (codimension 4 defects, just like instantons) can bind The kind of instanton counting we encountered occurs naturally in the theory of D4 branes in IIA string theory to which D0 branes (codimension 4 defects, just like instantons) can bind

M-theory to the rescue

D4 branes D0’s SU(4) rotation

M-theory to the rescue D4 brane+ D0’s become Lift to M-theory M5 brane wrapped on R 4 X elliptic curve Free fields = the tensor multiplet of (2,0) supersymmetry The modularity of the partition function is the consequence of the general covariance of the six dimensional theory NN+E.Witten

M-theory to the rescue In the limit To visualize this boson deform R 4 to Taub-Nut space The tensor field gets a normalizable localized mode The partition function becomes that of a free chiral boson on elliptic curve

Higher dimensional perspective on the gauge instanton counting Complicated hook measure on Partitions comes from simple Uniform measure on plane (3d) partitions

Higher dimensional perspective on the gauge instanton counting Complicated hook measure on Partitions comes from simple Uniform measure on plane (3d) partitions What is the physics of this relation?

Gauge theory = low energy limit of string theory compactification

X

Instanton partition function = String instanton partition function

Instanton partition function for gauge group G = String instanton partition function for special X Local CY’s Geometric enigneering Katz, Klemm, Vafa

Instanton partition function for gauge group G = String instanton partition function for special X Kontsevich’s moduli space of stable maps

String instanton partition function for CY X = counting holomorphic curves on X

String instanton partition function for CY X = counting holomorphic curves on X Gromov-Witten theory

Counting holomorphic curves on X (GW theory) = Counting equations describing holomorphic curves (ideal sheaves)

Counting equations describing holomorphic curves (ideal sheaves) Donaldson-Thomas theory

For special X, e.g. toric, Donaldson-Thomas theory can be done using localization = sum over fixed points = toric ideal sheaves

Simplest toric X = C 3 toric ideal sheaves = monomial ideals

Monomial ideals = three dimensional partitions

Topological vertex

Equivariant vertex (beyond CY)

K-theoretic Equivariant vertex (beyond string theory & CY)

The case of C 3 Contribution of a three dimensional partition Contribution of a three dimensional partition

The case of C 3 Contribution of a three dimensional partition Contribution of a three dimensional partition

The case of C 3 Contribution of a three dimensional partition Contribution of a three dimensional partition

The case of C 3 The partition function Counts bound states of D0’s and a D6 brane

The partition function has a free field realization

The partition function Special limits

If, in addition:

The partition function Special limits If, in addition: Our good old MacMahon friend

The partition function Second quantization

Explanation via M-theory Type IIA realization

Explanation via M-theory Lift to M-theory

Explanation via M-theory Deform TN to R 4 R 10 rotated over the circle: SU(5) rotation

Explanation via M-theory Free fields = linearized supergravity multiplet NN+E.Witten

Instanton partition functions  Generalize most known special functions (automorphic forms)  Obey interesting differential and difference equations  Relate combinatorics, algebra, representation theory and geometry; string theory and gauge theory  Might teach us about the nature of M-theory  Generalize most known special functions (automorphic forms)  Obey interesting differential and difference equations  Relate combinatorics, algebra, representation theory and geometry; string theory and gauge theory  Might teach us about the nature of M-theory