Skip to main content
SpringerLink
Log in

Gauge Fields and Interacting Particles

  • Chapter
Calogero—Moser— Sutherland Models

Part of the book series: CRM Series in Mathematical Physics ((CRM))

  • 720 Accesses

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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.

    Article  ADS  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  3. H. W. Braden and R. Sasaki, The Ruijsenaars-Schneider model, Progr. Theor. Phys. 97 (1997), No. 6, 1003–1017, hep-th/9702182.

    Article  MathSciNet  ADS  Google Scholar 

  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.

    Article  MathSciNet  ADS  Google Scholar 

  5. R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996), No. 2, 299–334.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. V. Fock, A. Gorsky, N. Nekrasov, and V. Rubtsov, Duality in integrable systems and gauge theories, hep-th/9906235.

    Google Scholar 

  7. A. Givental, Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices (1996), No. 13, 613–663, alg-geom/ 9603021.

    Google Scholar 

  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. 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.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. A. Gorsky and N. Nekrasov, Elliptic Calogero-Moser system from two-dimensional current algebra, hep-th/9401021.

    Google Scholar 

  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.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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.

    MathSciNet  ADS  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  ADS  Google Scholar 

  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. 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.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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. N. Nekrasov, On a duality in Calogero-Moser systems, Tech. Report ITEP-TH-16/97, ITEP, Moscow, 1997.

    Google Scholar 

  22. Five-dimensional gauge theories and relativistic integrable systems, Nucl. Phys. B 531 (1998), No. 1-3, 323–344, hep-th/9609219.

    Article  Google Scholar 

  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.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Classical integrable finite-dimensional systems related to Lie algebras, Phys. Rep. 71 (1981), No. 5, 313–400.

    Article  MathSciNet  Google Scholar 

  25. M. Rozali, Matrix theory and U-duality in seven dimensions, Phys. Lett. B 400 (1997), No. 3-4, 260–264, hep-th/9702136.

    Article  MathSciNet  ADS  Google Scholar 

  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.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Finite-dimensional soliton systems, Integrable and Superintegrable Systems (B. A. Kupershmidt, ed.), World Scientific, Singapore, 1990, pp. 165–206.

    Google Scholar 

  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.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994), No. 2, 484–550.

    Google Scholar 

  31. B. Sutherland, Exact results for a quantum many-body problem in one dimension. II, Phys. Rev. A 5 (1972), 1372–1376.

    Article  ADS  Google Scholar 

Download references

Authors
  1. N. Nekrasov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Departamento de Matematicas, Universidad de Chile, Las Palmeras, 3425, Nunoa Santiago, Chile

    Jan Felipe van Diejen

  2. McGill University, James Administration Building, Room 504, Montreal, Quebec, H3A 2T5, Canada

    Luc Vinet

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer Science+Business Media New York

About this chapter

Cite this chapter

Nekrasov, N. (2000). Gauge Fields and Interacting Particles. In: van Diejen, J.F., Vinet, L. (eds) Calogero—Moser— Sutherland Models. CRM Series in Mathematical Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1206-5_23

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-1206-5_23

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7043-0

  • Online ISBN: 978-1-4612-1206-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation