On the convergence of Nekrasov functions

TL;DR

This paper investigates the convergence properties of Nekrasov partition functions in four-dimensional N=2 gauge theories, establishing bounds on their convergence radius and implications for related conformal blocks and Painlevé tau-functions.

Contribution

It provides rigorous bounds on the convergence radius of Nekrasov functions for conformal and asymptotically free theories, linking gauge theory, conformal blocks, and Painlevé equations.

Findings

Conformal theories have at least a finite convergence radius.

Asymptotically free theories have infinite convergence radius.

Results imply convergence of related conformal blocks and Painlevé tau-functions.

Abstract

In this note we present some results on the convergence of Nekrasov partition functions as power series in the instanton counting parameter. We focus on $U(N)$ ${\mathcal N}=2$ gauge theories in four dimensions with matter in the adjoint and in the fundamental representations of the gauge group respectively and find rigorous lower bounds for the convergence radius in the two cases: if the theory is {\it conformal}, then the series has at least a {\it finite} radius of convergence, while if it is {\it asymptotically free} it has {\it infinite} radius of convergence. Via AGT correspondence, this implies that the related irregular conformal blocks of $W_N$ algebrae admit a power expansion in the modulus converging in the whole plane. By specifying to the $SU(2)$ case, we apply our results to analyse the convergence properties of the corresponding Painlev\'e $\tau$-functions.