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

TL;DR

The paper proves Nekrasov's conjecture by showing that equivariant integration over instanton moduli spaces on ${f R}^4$ yields a deformation of the Seiberg–Witten prepotential. It develops a blowup formula on the framed moduli spaces over the blowup ${f ilde{C}}^2$, introduces a $oldsymbolullet(C)$ operator, and derives a differential blowup equation that recursively determines the instanton part $F^{ ext{inst}}$; in the limit $oldsymboleta o0$, the instanton part matches the Seiberg–Witten prepotential, establishing the conjecture. The approach also connects Nekrasov's partition function to Hilbert-series via equivariant K-theory, and extends the framework to general gauge groups, highlighting a deep bridge between localization, blowup geometry, and Seiberg–Witten theory.

Abstract

We give a mathematically rigorous proof of Nekrasov's conjecture: the integration in the equivariant cohomology over the moduli spaces of instantons on $\mathbb R^4$ 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 $\mathbb R^4$, 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.

Figures (1)

• Figure 1: Young diagram and ideal

Theorems & Definitions (45)

• Remark 1.6
• Proposition 2.1
• proof
• Corollary 2.2
• proof
• Remark 2.6
• Lemma 2.8
• proof
• Proposition 2.9
• proof