Seiberg-Witten Prepotential from Instanton Counting

Abstract

In my lecture I consider integrals over moduli spaces of supersymmetric gauge field configurations (instantons, Higgs bundles, torsion free sheaves ).The applications are twofold: physical and mathematical; they involve supersymmetric quantum mechanics of D-particles in various dimensions, direct computation of the celebrated Seiberg-Witten prepotential, sum rules for the solutions of the Bethe ansatz equations and their relation to the Laumon's nilpotent cone. As a by-product we derive some combinatoric identities involving the sums over Young tableaux.

Gravitational corrections in supersymmetric gauge theory and matrix models

“…Here we provide nontrivial evidence for this conjecture by comparing some of the instanton computations of [10,11] with the results of section 2 (and, therefore, with the matrix model computations). 5 Note that the scale appearing in sect.…”

“…In a recent tour de force Nekrasov [10](see also [11]) was able to provide general expressions for the n-th instanton contribution to the N = 2, SU (N ) prepotential, with or without matter. The answer for the n-th instanton correction has the form F n (a, 1 , 2 ) and it is an analytic function in 1,2 .…”

“…For 1 = − 2 = , one can expand this as The first coefficient in this expansion, F (0) n (a), gives the prepotential of the corresponding N = 2 theory. It was conjectured in [10] that the remaining coefficients F (g) n (a), g ≥ 1, give the n-instanton correction to the gravitational couplings F (g) of the N = 2 theory.…”

“…For N = 2 theories, the genus one coupling can be extracted from the low-energy twisted theory, where the coupling to gravity plays an important role [6,7,8,9]. More recently there has been important progress in calculating also the space-time instantons contributions for higher genus [10,11].…”

Restriction theorems forμ-(semi)stable framed sheaves

“…Because of the relation with moduli space of framed instantons, since Nekrasov's partition function was introduced in [22], the moduli space M(r, n) has been studied quite intensively (see, e.g., [1,19,20,21,4]) and the geometry of moduli spaces of framed sheaves on the complex projective plane is quite well known.…”