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INSTANTON PARTITION FUNCTIONS

INSTANTON PARTITION FUNCTIONS. 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;

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INSTANTON PARTITION FUNCTIONS

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  1. INSTANTON PARTITION FUNCTIONS 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

  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) …

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

  7. Mathematical problem:generating functions

  8. Mathematical problem:generating functions

  9. Mathematical problem:generating functions 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));….

  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));….

  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) Intersection theory on the moduli space of gauge instantons

  29. Instanton partition functions in four dimensions 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 R4

  31. Instanton partition functions in four dimensions Supersymmetric N=2 theory On Euclidean space R4 Boundary conditions at infinity SO(4) Equivariant theory

  32. Instanton partition function Supersymmetric N=2 theory on Euclidean space R4

  33. Instanton partition function Supersymmetric pure N=2 super YM theory on Euclidean space R4 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. Instanton partition function 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

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