Instanton counting on blowup. II. $K$-theoretic partition function

TL;DR

This paper explores the K-theoretic partition function of 5D supersymmetric Yang-Mills theory, establishing functional equations and deriving explicit formulas for genus 1 terms related to Seiberg-Witten curves.

Contribution

It introduces blowup equations for the K-theoretic partition function and proves their uniqueness, advancing understanding of instanton moduli spaces and Nekrasov's conjecture.

Findings

Partition function satisfies unique blowup equations

Logarithm of the partition function is regular at zero

Explicit formulas for genus 1 terms in rank 2 case

Abstract

We study Nekrasov's deformed partition function of 5-dimensional supersymmetric Yang-Mills theory compactified on a circle. Mathematically it is the generating function of the characters of the coordinate rings of the moduli spaces of instantons on $\mathbb R^4$. We show that it satisfies a system of functional equations, called blowup equations, whose solution is unique. As applications, we prove (a) logarithm of the partition function times $\epsilon_1\epsilon_2$ is regular at $\epsilon_1 = \epsilon_2 = 0$, (a part of Nekrasov's conjecture), and (b) the genus 1 parts, which are first several Taylor coefficients of the logarithm of the partition function, are written explicitly in terms of the Seiberg-Witten curves in rank 2 case.