Two Dimensional Gauge Theories and Quantum Integrable Systems

1. Two Dimensional Gauge Theoriesand Quantum Integrable Systems Nikita Nekrasov IHES Imperial College April 10, 2008

2. Based on NN, S.Shatashvili, to appear Prior work: E.Witten, 1992; A.Gorsky, NN; J.Minahan, A.Polychronakos; M.Douglas; ~1993-1994; A.Gerasimov ~1993; G.Moore, NN, S.Shatashvili ~1997-1998; A.Losev, NN, S.Shatashvili ~1997-1998; A.Gerasimov, S.Shatashvili ~ 2006-2007

3. We are going to relate 2,3, and 4 dimensional susy gauge theorieswith four supersymmetries N=1 d=4 And quantum integrable systems soluble by Bethe Ansatz techniques.

4. Mathematically speaking, the cohomology, K-theory and elliptic cohomology of various gauge theory moduli spaces, like moduli of flat connections and instantons And quantum integrable systems soluble by Bethe Ansatz techniques.

5. For example, we shall relate the XXX Heisenberg magnet and 2d N=2 SYM theory with some matter

6. (pre-)History In 1992 E.Witten studied two dimensional Yang-Mills theory with the goal to understand the relation between the physical and topological gravities in 2d.

7. (pre-)History There are two interesting kinds of Two dimensional Yang-Mills theories

8. Yang-Mills theories in 2d (1) Cohomological YM = twisted N=2 super-Yang-Mills theory, with gauge group G, whose BPS (or TFT) sector is related to the intersection theory on the moduli space MG of flat G-connections on a Riemann surface

9. Yang-Mills theories in 2d N=2 super-Yang-Mills theory Field content:

10. Yang-Mills theories in 2d (2) Physical YM = N=0 Yang-Mills theory, with gauge group G; The moduli space MG of flat G-connections = minima of the action; The theory is exactly soluble (A.Migdal) with the help of the Polyakov lattice YM action

11. Yang-Mills theories in 2d Physical YM Field content:

12. Yang-Mills theories in 2d Witten found a way to map the BPS sector of the N=2 theory to the N=0 theory. The result is:

13. Yang-Mills theories in 2d Two dimensional Yang-Mills partition function is given by the explicit sum

14. Yang-Mills theories in 2d In the limit the partition function computes the volume of MG

15. Yang-Mills theories in 2d Witten’s approach: add twisted superpotential and its conjugate

16. Yang-Mills theories in 2d Take a limit In the limit the fields are infinitely massive and can be integrated out: one is left with the field content of the physical YM theory

17. Yang-Mills theories in 2d Both physical and cohomological Yang-Mills theories define topological field theories (TFT)

18. Yang-Mills theories in 2d Both physical and cohomological Yang-Mills theories define topological field theories (TFT) Vacuum states + deformations = quantum mechanics

19. YM in 2d and particles on a circle Physical YM is explicitly equivalent to a quantum mechanical model: free fermions on a circle Can be checked by a partition function on a two-torus Gross Douglas

20. YM in 2d and particles on a circle Physical YM is explicitly equivalent to a quantum mechanical model: free fermions on a circle States are labelled by the partitions, for G=U(N)

21. YM in 2d and particles on a circle For N=2 YM these free fermions on a circle Label the vacua of the theory deformed by twisted superpotential W

22. YM in 2d and particles on a circle The fermions can be made interacting by adding a localized matter: for example a time-like Wilson loop in some representation V of the gauge group:

23. YM in 2d and particles on a circle One gets Calogero-Sutherland (spin) particles on a circle (1993-94) A.Gorsky,NN; J.Minahan,A.Polychronakos;

24. History In 1997 G.Moore, NN and S.Shatashvili studied integrals over various hyperkahler quotients, with the aim to understand instanton integrals in four dimensional gauge theories

25. History In 1997 G.Moore, NN and S.Shatashvili studied integrals over various hyperkahler quotients, with the aim to understand instanton integrals in four dimensional gauge theories This eventually led to the derivation in 2002 of the Seiberg-Witten solution of N=2 d=4 theory Inspired by the work of H.Nakajima

26. Yang-Mills-Higgs theory Among various examples, MNS studied Hitchin’s moduli space MH

27. Yang-Mills-Higgs theory Unlike the case of two-dimensional Yang-Mills theory where the moduli space MG is compact, Hitchin’s moduli space is non-compact (it is roughly T*MG modulo subtleties) and the volume is infinite.

28. Yang-Mills-Higgs theory In order to cure this infnity in a reasonable way MNS used the U(1) symmetry of MH The volume becomes a DH-type expression: Where H is the Hamiltonian

29. Yang-Mills-Higgs theory Using the supersymmetry and localization the regularized volume ofMH was computed with the result

30. Yang-Mills-Higgs theory Where the eigenvalues solve the equations:

31. YMH and NLS The experts would immediately recognise the Bethe ansatz (BA) equations for the non-linear Schroedinger theory (NLS) NLS = large spin limit of the SU(2) XXX spin chain

32. YMH and NLS Moreover the NLS Hamiltonians are the 0-observables of the theory, like The VEV of the observable = The eigenvalue of the Hamiltonian

33. YMH and NLS Since 1997 nothing came out of this result. It could have been simply a coincidence. …….

34. In 2006 A.Gerasimov and S.Shatashvili have revived the subject History

35. YMH and interacting particles GS noticed that YMH theory viewed as TFT is equivalent to the quantum Yang system: N particles on a circle with delta-interaction:

36. YMH and interacting particles Thus: YM with the matter -- fermions with pair-wise interaction

37. History More importantly, GS suggested that TFT/QIS equivalence is much more universal

38. Today We shall rederive the result of MNS from a modern perspective Generalize to cover virtually all BA soluble systems both with finite and infinite spin Suggest natural extensions of the BA equations

39. Hitchin equations Solutions can be viewed as the susy field configurations for the N=2 gauged linear sigma model For adjoint-valued linear fields

40. Hitchin equations The moduli space MH of solutions is a hyperkahler manifold The integrals over MH are computed by the correlation functions of an N=2 d=2 susy gauge theory

41. Hitchin equations The kahler form on MH comes from twisted tree level superpotential The epsilon-term comes from a twisted mass of the matter multiplet

42. Generalization Take an N=2 d=2 gauge theory with matter, In some representation R of the gauge group G

43. Generalization Integrate out the matter fields, compute the effective (twisted) super-potential on the Coulomb branch

44. Mathematically speaking Consider the moduli space MR of R-Higgs pairs with gauge group G Up to the action of the complexified gauge group GC

45. Mathematically speaking Stability conditions: Up to the action of the compact gauge group G

46. Mathematically speaking Pushforward the unit class down to the moduli space MG of GC-bundles Equivariantly with respect to the action of the global symmetry group K on MR

47. Mathematically speaking The pushforward can be expressed in terms of the Donaldson-like classes of the moduli space MG 2-observables and 0-observables

48. Mathematically speaking The pushforward can be expressed in terms of the Donaldson-like classes of the moduli space MG 2-observables and 0-observables

49. Mathematically speaking The masses are the equivariant parameters For the global symmetry group K

50. Vacua of the gauge theory Due to quantization of the gauge flux For G = U(N)