Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 008, 11 pages      hep-th/0601167      http://dx.doi.org/10.3842/SIGMA.2006.008

Status Report on the Instanton Counting

Sergey Shadchin
INFN, Sezione di Padova & Dipartimento di Fisica “G. Galilei”, Università degli Studi di Padova, via F. Marzolo 8, Padova, 35131, Italy

Received December 07, 2005, in final form January 18, 2006; Published online January 22, 2006

Abstract
The non-perturbative behavior of the N = 2 supersymmetric Yang-Mills theories is both highly non-trivial and tractable. In the last three years the valuable progress was achieved in the instanton counting, the direct evaluation of the low-energy effective Wilsonian action of the theory. The localization technique together with the Lorentz deformation of the action provides an elegant way to reduce functional integrals, representing the effective action, to some finite dimensional contour integrals. These integrals, in their turn, can be converted into some difference equations which define the Seiberg-Witten curves, the main ingredient of another approach to the non-perturbative computations in the N = 2 super Yang-Mills theories. Almost all models with classical gauge groups, allowed by the asymptotic freedom condition can be treated in such a way. In my talk I explain the localization approach to the problem, its relation to the Seiberg-Witten approach and finally I give a review of some interesting results.

Key words: instanton counting; Seiberg-Witten theory.

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References

  1. Baulieu L., Singer I.M., Topological Yang-Mills symmetry, Nucl. Phys. B, Proc. Suppl., 1988, V.5, 12-19.
  2. Duistermaat J.J., Heckman G.J., On the variation in the cohomology in the symplectic form of the reduced phase space, Invent. Math., 1982, V.69, 259-269.
  3. Erlich J., Naqvi A., Randall L., The Coulomb branch of N = 2 supersymmetric product group theories from branes, Phys. Rev. D, 1998, V.58, 046002, 10 pages, hep-th/9801108.
  4. Katz S., Mayr P., Vafa C., Mirror symmetry and exact solution of 4d n = 2 gauge theories. I, Adv. Theor. Math. Phys., 1998, V.1, 53-114, hep-th/9706110.
  5. Katz S., Klemm A., Vafa C., Geometric engineering of quantum field theories, Nucl. Phys. B, 1997, V.497, 173-195, hep-th/9609239.
  6. Losev A., Marshakov A., Nekrasov N., Small instantons, little strings and free fermions, hep-th/0302191.
  7. Mariño M., Wyllard N., A note on instanton counting for N = 2 gauge theories with classical gauge groups, JHEP, 2004, V.0405, paper 021, 23 pages, hep-th/0404125.
  8. Nekrasov N., Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys., 2004, V.7, 831-864, hep-th/0206161.
  9. Nekrasov N., Okounkov A., Seiberg-Witten theory and random partitions, hep-th/0306238.
  10. Nekrasov N., Shadchin S., ABCD of instantons, Comm. Math. Phys., 2004, V.253, 359-391, hep-th/0404225.
  11. Seiberg N., Supersymmetry and nonperturbative beta functions, Phys. Lett. B, 1988, V.206, 75-87.
  12. Seiberg N., Witten E., Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B, 1994, V.426, 19-52, hep-th/9407087.
  13. Seiberg N., Witten E., Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD, Nucl. Phys. B, 1994, V.431, 484-550, hep-th/9408099.
  14. Shadchin S., Cubic curves from instanton counting, hep-th/0511132.
  15. Shadchin S., On certain aspects of string theory/gauge theory correspondence, PhD Thesis, Université Paris-Sud, Orsay, France, 2005, hep-th/0502180.
  16. Shadchin S., Saddle point equations in Seiberg-Witten theory, JHEP, 2004, V.0410, paper 033, 38 pages, hep-th/0408066.
  17. Witten E., Introduction to topological quantum field theories, Lectures at the Workshop on Topological Methods in Physics, ICTP, Trieste, Italy (June 1990).
  18. Witten E., Topological quantum field theory, Comm. Math. Phys., 1988, V.117, 353-386.
  19. Witten E., Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B, 1997, V.500, 3-42, hep-th/9703166.
  20. Witten E., Donagi R., Supersymmetric Yang-Mills systems and integrable systems, Nucl. Phys. B, 1996, V.460, 299-334, hep-th/9510101.

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