Perverse coherent sheaves on blow-up. III. Blow-up formula from wall-crossing

TL;DR

This paper investigates how Donaldson invariants change across a sequence of moduli spaces of perverse coherent sheaves on blow-ups, deriving recursive equations for Nekrasov-type partition functions related to supersymmetric gauge theories.

Contribution

It establishes a blow-up formula for Donaldson invariants via wall-crossing of moduli spaces of perverse coherent sheaves, extending Nekrasov partition function equations.

Findings

Derived recursive equations for Nekrasov-type partition functions.

Proved invariance relations for Donaldson invariants under wall-crossing.

Connected mathematical invariants with physical gauge theory models.

Abstract

In earlier papers arXiv:0802.3120, arXiv:0806.0463 of this series we constructed a sequence of intermediate moduli spaces $\bM^m(c)$ connecting a moduli space $M(c)$ of stable torsion free sheaves on a nonsingular complex projective surface and $\bM(c)$ on its one point blow-up. They are moduli spaces of perverse coherent sheaves on the blow-up. In this paper we study how Donaldson-type invariants (integrals of cohomology classes given by universal sheaves) change from $\bM^m(c)$ to $\bM^{m+1}(c)$, and then from $M(c)$ to $\bM(c)$. As an application we prove that Nekrasov-type partition functions satisfy certain equations which determine invariants recursively in second Chern classes. They are generalization of the blow-up equation for the original Nekrasov deformed partition function for the pure N=2 SUSY gauge theory, found and used to derive the Seiberg-Witten curves in arXiv:math/0306198.