Gauge Fields and Interacting Particles

  • N. Nekrasov
Part of the CRM Series in Mathematical Physics book series (CRM)


In this short survey we discuss integrable many-body systems and their relation to gauge theories. One aspect of such a relation is the Hamiltonian reduction, which produces the model out of a simple dynamical system. The phase spaces of original simple systems are constructed using the infinite-dimensional current algebras. We briefly discuss dualities, relating integrable systems. Finally, we outline the applications to the studies of moduli spaces of vacua of supersymmetric gauge theories in four and five dimensions and present derivations of some of these results using string theory.


Gauge Theory Modulus Space Vector Multiplet Supersymmetric Gauge Theory Hamiltonian Reduction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    P. C. Argyres and A. E. Farragi, The vacuum structure and spectrum of N = 2 super symmetric SU(n) gauge theory, Phys. Rev. Lett. 74 (1995), 3931–3934, hep-th/9411057.ADSCrossRefGoogle Scholar
  2. 2.
    V. V. Batyrev, Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori, Duke Math. J. 69 (1993), No. 2, 349–409.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    H. W. Braden and R. Sasaki, The Ruijsenaars-Schneider model, Progr. Theor. Phys. 97 (1997), No. 6, 1003–1017, hep-th/9702182.MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    F. Calogero, Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971), 419–436.MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996), No. 2, 299–334.MathSciNetADSMATHCrossRefGoogle Scholar
  6. 6.
    V. Fock, A. Gorsky, N. Nekrasov, and V. Rubtsov, Duality in integrable systems and gauge theories, hep-th/9906235.Google Scholar
  7. 7.
    A. Givental, Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices (1996), No. 13, 613–663, alg-geom/ 9603021.Google Scholar
  8. 8.
    A mirror theorem for complete intersections, Topological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996) (M. Kashiwara, A. Matsuo, K. Saito, and I. Satake, eds.), Progr. Math., Vol. 160, Birkhäuser, Boston, MA, 1998, pp. 141–175, alg-geom/9701016.Google Scholar
  9. 9.
    A. Gorsky, I. M. Krichever, A. Marshakov, A. Morozov, and A. Mironov, Integrablity and Seiberg-Witten exact solution, Phys. Lett. B 355 (1995), No. 3-4, 466–474, hep-th/9505035.MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    A. Gorsky and N. Nekrasov, Elliptic Calogero-Moser system from two-dimensional current algebra, hep-th/9401021.Google Scholar
  11. 11.
    A. Gorsky and N. Nekrasov, Hamiltonian systems of Calogero-type, and two-dimensional Yang-Mills theory, Nucl. Phys. B 414 (1994), No. 1-2, 213–238.MathSciNetADSMATHCrossRefGoogle Scholar
  12. 12.
    A. Gorsky and N. Nekrasov, Relativistic Calogero-Moser model as gauged WZW theory, Nucl. Phys. B 436 (1995), No. 3, 582–608, hep-th/9401017.MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles, and three-dimensional gauge theories, Nucl. Phys. B 492 (1997), No. 1-2, 152–190, hep-th/9611230.MathSciNetADSGoogle Scholar
  14. 14.
    S. Katz, P. Mayr, and C. Vafa, Mirror symmetry and exact solution of 4D N = 2 gauge theories. I, Adv. Theor. Math. Phys. 1 (1997), No. 1, 53–114, hep-th/9706110.MathSciNetMATHGoogle Scholar
  15. 15.
    D. Kazhdan, B. Kostant, and S. Sternberg, Hamiltonian group actions and dynamical systems of Calogero type, Commun. Pure Appi. Math. 31 (1978), No. 4, 481–507.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    A. Klemm, W. Lerche, S. Theisen, and S. Yankielowicz, Simple singularities and N = 2 supersymmetric Yang-Mills theory, Phys. Lett. B 344 (1995), No. 1-4, 169–175, hep-th/9411048.MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    M. Kontsevich, Enumeration of rational curves via torus actions, The Moduli Space of Curves (Texel, 1994) (R. Dijkgraaf, C. Faber, and G. van der Geer, eds.), Progr. Math., Vol. 129, Birkhäuser, Boston, MA, 1995, pp. 335–368, hep-th/9405035.Google Scholar
  18. 18.
    A. Lawrence and N. Nekrasov, Instanton sums and five-dimensional gauge theories, Nucl. Phys. B 513 (1998), No. 1-2, 239–265, hep-th/9706025.MathSciNetADSMATHCrossRefGoogle Scholar
  19. 19.
    D. R. Morrison and M. R. Plesser, Summing up the instantons: quantum cohomology and mirror symmetry in torio varieties, Nucl. Phys. B 440 (1995), No. 1-2, 279–354, hep-th/9412136.MathSciNetADSMATHCrossRefGoogle Scholar
  20. 20.
    Towards mirror symmetry as duality for two-dimensional Abelian gauge theories, S-duality and Mirror Symmetry (Trieste, 1995) (E. Gava, K. S. Narain, and C. Vafa, eds.), Nuclear Phys. B Proc. Suppl., Vol. 46, North-Holland, Amsterdam, 1997, pp. 177–186, hep-th/9508107.Google Scholar
  21. 21.
    N. Nekrasov, On a duality in Calogero-Moser systems, Tech. Report ITEP-TH-16/97, ITEP, Moscow, 1997.Google Scholar
  22. 22.
    Five-dimensional gauge theories and relativistic integrable systems, Nucl. Phys. B 531 (1998), No. 1-3, 323–344, hep-th/9609219.CrossRefGoogle Scholar
  23. 23.
    M. A. Olshanetsky and A. M. Perelomov, Completely integrable Hamiltonian systems connected with semisimple Lie algebras, Invent. Math. 37 (1976), No. 2, 93–108.MathSciNetADSMATHCrossRefGoogle Scholar
  24. 24.
    Classical integrable finite-dimensional systems related to Lie algebras, Phys. Rep. 71 (1981), No. 5, 313–400.MathSciNetCrossRefGoogle Scholar
  25. 25.
    M. Rozali, Matrix theory and U-duality in seven dimensions, Phys. Lett. B 400 (1997), No. 3-4, 260–264, hep-th/9702136.MathSciNetADSCrossRefGoogle Scholar
  26. 26.
    S. N. M. Ruijsenaars, Complete integrability of relativistic Calogero-Moser systems and elliptic function identities, Commun. Math. Phys. 110 (1987), No. 2, 191–213.MathSciNetADSMATHCrossRefGoogle Scholar
  27. 27.
    Finite-dimensional soliton systems, Integrable and Superintegrable Systems (B. A. Kupershmidt, ed.), World Scientific, Singapore, 1990, pp. 165–206.Google Scholar
  28. 28.
    S. N. M. Ruijsenaars and H. Schneider, A new class of integrable systems and its relation to solitons, Ann. Phys. (NY) 170 (1986), No. 2, 370–405.MathSciNetADSMATHCrossRefGoogle Scholar
  29. 29.
    N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994), No. 1, 19–52, Errata, 430 (1994), no. 2, 485-486.MathSciNetADSMATHCrossRefGoogle Scholar
  30. 30.
    Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994), No. 2, 484–550.Google Scholar
  31. 31.
    B. Sutherland, Exact results for a quantum many-body problem in one dimension. II, Phys. Rev. A 5 (1972), 1372–1376.ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • N. Nekrasov

There are no affiliations available

Personalised recommendations