1. 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

2. 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 ~1997-1998; A.Losev, NN, S.Shatashvili ~1997-1998; A.Gerasimov, S.Shatashvili ~ 2006-2007 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 ~1997-1998; A.Losev, NN, S.Shatashvili ~1997-1998; A.Gerasimov, S.Shatashvili ~ 2006-2007

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

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

5. 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) …

6. 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) …

7. Mathematical problem: generating functions

9. Euler

10. Unexpected symmetry Dedekind eta

11. More structure: Arms, legs, and hooks

12. Growth process

13. Plancherel measure

14. 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));….

15. 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));….

16. Mathematical problem: generating functions MacMahon

17. Mathematical problem: more structural counting

18. Quantum gauge theory Four dimensions

19. Quantum gauge theory Four dimensions

20. Quantum sigma model Two dimensions

21. Quantum sigma model Two dimensions

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

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

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

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

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

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

28. 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

29. 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

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

31. 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

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

33. 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

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

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

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

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

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

39. Instanton partition function

40. For G = U(N)

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

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

43. Instanton partition function Generalized growth model

44. Instanton partition function Generalized growth model

45. Instanton partition function Generalized growth model

46. Instanton partition function Generalized growth model

47. Instanton partition function Generalized growth model

48. Instanton partition function Generalized growth model

49. Instanton partition function Limit shape Emerging geometry

50. Instanton partition function Limit shape Emerging algebraic geometry

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

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

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

54. 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

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

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

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

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

59. Instanton partition function Free fermion representation

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

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

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

63. 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

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

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

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

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

68. Instanton partition function Free fields? Where? What kind?

69. 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

70. M-theory to the rescue

71. D4 branes D0’s SU(4) rotation

72. 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

73. 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

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

75. 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?

76. Gauge theory = low energy limit of string theory compactification

77. X

78. Instanton partition function = String instanton partition function

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

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

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

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

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

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

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

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

88. Monomial ideals = three dimensional partitions

90. Topological vertex

91. Equivariant vertex (beyond CY)

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

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

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

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

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

97. The partition function has a free field realization

98. The partition function Special limits

99. If, in addition:

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

101. The partition function Second quantization

102. Explanation via M-theory Type IIA realization

103. Explanation via M-theory Lift to M-theory

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

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

106. 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