The Nekrasov Conjecture for Toric Surfaces

TL;DR

This paper proves a generalized version of the Nekrasov conjecture, linking the partition function of supersymmetric gauge theories to geometric invariants on noncompact toric surfaces, extending previous results on R^4.

Contribution

It extends the Nekrasov conjecture proof from R^4 to noncompact toric surfaces, broadening the mathematical understanding of gauge theory invariants.

Findings

Confirmed the conjecture for a class of noncompact toric surfaces

Established new relations between partition functions and geometric invariants

Extended the mathematical framework for supersymmetric gauge theories

Abstract

The Nekrasov conjecture predicts a relation between the partition function for N=2 supersymmetric Yang-Mills theory and the Seiberg-Witten prepotential. For instantons on R^4, the conjecture was proved, independently and using different methods, by Nekrasov-Okounkov, Nakajima-Yoshioka, and Braverman-Etingof. We prove a generalized version of the conjecture for instantons on noncompact toric surfaces.