Analyticity of Nekrasov Partition Functions

TL;DR

This paper proves that K-theoretic Nekrasov instanton partition functions are holomorphic with a positive radius of convergence, impacting the understanding of the AGT correspondence and related algebraic structures.

Contribution

It establishes the analyticity and convergence properties of Nekrasov partition functions using random matrix techniques and integral representations.

Findings

Partition functions have a positive radius of convergence.

Partition functions are holomorphic in Coulomb parameters.

Implications for AGT correspondence and Gaiotto states.

Abstract

We prove that the K-theoretic Nekrasov instanton partition functions have a positive radius of convergence in the instanton counting parameter and are holomorphic functions of the Coulomb parameters in a suitable domain. We discuss the implications for the AGT correspondence and the analyticity of the norm of Gaiotto states for the deformed Virasoro algebra. The proof is based on random matrix techniques and relies on an integral representation of the partition function, due to Nekrasov, which we also prove.