
SIGMA 2 (2006), 008, 11 pages hepth/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 nonperturbative behavior of the
N = 2 supersymmetric YangMills theories is both
highly nontrivial and tractable. In the last three years
the valuable progress was achieved in the instanton counting,
the direct evaluation of the lowenergy 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 SeibergWitten
curves, the main ingredient of another approach to the nonperturbative
computations in the N = 2 super YangMills 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 SeibergWitten
approach and finally I give a review of some interesting results.
Key words:
instanton counting; SeibergWitten theory.
pdf (252 kb)
ps (215 kb)
tex (32 kb)
References
 Baulieu L., Singer I.M., Topological YangMills symmetry,
Nucl. Phys. B, Proc. Suppl., 1988, V.5, 1219.
 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, 259269.
 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,
hepth/9801108.
 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, 53114,
hepth/9706110.
 Katz S., Klemm A., Vafa C., Geometric engineering of quantum
field theories, Nucl. Phys. B, 1997, V.497, 173195, hepth/9609239.
 Losev A., Marshakov A., Nekrasov N., Small instantons, little strings and
free fermions, hepth/0302191.
 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, hepth/0404125.
 Nekrasov N., SeibergWitten prepotential from instanton counting,
Adv. Theor. Math. Phys., 2004, V.7, 831864,
hepth/0206161.
 Nekrasov N., Okounkov A., SeibergWitten theory and random partitions,
hepth/0306238.
 Nekrasov N., Shadchin S., ABCD of instantons,
Comm. Math. Phys., 2004, V.253, 359391, hepth/0404225.
 Seiberg N., Supersymmetry and nonperturbative beta functions, Phys. Lett. B, 1988, V.206, 7587.
 Seiberg N., Witten E., Electricmagnetic duality, monopole condensation, and
confinement in N = 2 supersymmetric YangMills theory,
Nucl. Phys. B, 1994, V.426, 1952, hepth/9407087.
 Seiberg N., Witten E., Monopoles, duality and chiral symmetry breaking in N=2
supersymmetric QCD, Nucl. Phys. B, 1994, V.431, 484550, hepth/9408099.
 Shadchin S., Cubic curves from instanton counting,
hepth/0511132.
 Shadchin S., On certain aspects of string theory/gauge theory correspondence, PhD Thesis,
Université ParisSud, Orsay, France, 2005,
hepth/0502180.
 Shadchin S., Saddle point equations in SeibergWitten theory,
JHEP, 2004, V.0410, paper 033, 38 pages, hepth/0408066.
 Witten E., Introduction to topological quantum field theories, Lectures at the
Workshop on Topological Methods in Physics, ICTP, Trieste, Italy (June 1990).
 Witten E., Topological quantum field theory, Comm. Math. Phys., 1988,
V.117, 353386.
 Witten E., Solutions of fourdimensional field theories via Mtheory,
Nucl. Phys. B, 1997, V.500, 342, hepth/9703166.
 Witten E., Donagi R., Supersymmetric YangMills systems and integrable
systems, Nucl. Phys. B, 1996, V.460, 299334, hepth/9510101.

