Instanton counting on blowup. I. 4-dimensional pure gauge theory

Abstract

Abstract. We give a mathematically rigorous proof of Nekrasov's conjecture: the integration in the equivariant cohomology over the moduli spaces of instantons on R 4 gives a deformation of the Seiberg-Witten prepotential for N = 2 SUSY Yang-Mills theory. Through a study of moduli spaces on the blowup of R 4 , we derive a differential equation for the Nekrasov's partition function. It is a deformation of the equation for the Seiberg-Witten prepotential, found by Losev et al., and further studied by Gorsky et al.

Vertex operators and the geometry of moduli spaces of framed torsion-free sheaves

“…Over the C * -fixed point (E 1 ⊕ E 2 , Φ 1 ⊕ Φ 2 ), the complex (3.4) also decomposes into isotypic components. As in the proof of [20,Theorem 2.11], ones see that the complex (3.1) is the isotypic subcomplex of (3.4) of weight 1. Similarly the subbundle Ext 1 (E 1 , E 2 (−l ∞ )) is the isotypic subbundle of…”

“…Denote the size of the partition λ by |λ| = i λ i . Following [20], we will identify a partition λ with the set…”

“…. , I r ), with c(I) = c and |λ(I)| = n (see [20,Proposition 2.9]). We shall identify T -fixed points with such r-tuples in what follows.…”

“…ford operators Using arguments similar to those appearing in [20,Section 2], one can interpret our geometric Heisenberg and Clifford operators in the language of sheaves. In so doing, we see that in the rank 1 case they are closely related to operators defined by Carsslon-Okouknov in [2].…”

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.…”