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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

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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 ~

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

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

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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) …

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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) …

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Mathematical problem: generating functions

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Euler

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Unexpected symmetry Dedekind eta

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More structure: Arms, legs, and hooks

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Growth process

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Plancherel measure

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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));….

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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));….

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Mathematical problem: generating functions MacMahon

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Mathematical problem: more structural counting

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Quantum gauge theory Four dimensions

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Quantum gauge theory Four dimensions

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Quantum sigma model Two dimensions

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Quantum sigma model Two dimensions

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

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

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

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

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

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

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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

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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

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

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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

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

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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

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

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

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

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

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

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Instanton partition function

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For G = U(N)

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

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

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Instanton partition function Generalized growth model

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Instanton partition function Generalized growth model

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Instanton partition function Generalized growth model

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Instanton partition function Generalized growth model

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Instanton partition function Generalized growth model

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Instanton partition function Generalized growth model

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Instanton partition function Limit shape Emerging geometry

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Instanton partition function Limit shape Emerging algebraic geometry

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

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

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

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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

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

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

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

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

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Instanton partition function Free fermion representation

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

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

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

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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

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

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

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

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

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Instanton partition function Free fields? Where? What kind?

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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

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M-theory to the rescue

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D4 branes D0’s SU(4) rotation

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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

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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

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

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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?

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Gauge theory = low energy limit of string theory compactification

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X

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Instanton partition function = String instanton partition function

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

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

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

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

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

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

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

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

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Monomial ideals = three dimensional partitions

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Topological vertex

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Equivariant vertex (beyond CY)

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

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

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

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

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

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The partition function has a free field realization

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The partition function Special limits

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If, in addition:

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

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The partition function Second quantization

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Explanation via M-theory Type IIA realization

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Explanation via M-theory Lift to M-theory

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

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

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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

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