Download presentation

Presentation is loading. Please wait.

Published byRayna Durrell Modified over 2 years ago

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

Similar presentations

Presentation is loading. Please wait....

OK

Seiberg Duality James Barnard University of Durham.

Seiberg Duality James Barnard University of Durham.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Dsp ppt on dft Ppt on human chromosomes 46 Ppt on disk formatting in windows Ppt on credit default swaps history Small intestine anatomy and physiology ppt on cells Ppt on recent natural disasters in india Ppt on cartesian product of graphs Ppt on file system in unix what is domain Ppt on windows 8 operating system download Ppt on types of chemical bonding