Convergence of Nekrasov instanton sum with adjoint matter

TL;DR

This paper proves the convergence of the Nekrasov instanton sum for 4d $ ext{N}=2^*$ $U(N)$ gauge theory within the unit disk, analyzing how convergence depends on parameters and relating results to conformal blocks via AGT correspondence.

Contribution

It establishes the conditions for convergence of the Nekrasov instanton series, including explicit radius calculations and parameter restrictions, extending understanding of instanton sums and their analytic properties.

Findings

Convergence radius is 1 for generic parameters with non-real $b$.

Series diverges for super-exponentially well approximable $b^2$.

Rational $b^2$ lead to singular terms in the series.

Abstract

The Nekrasov instanton partition function of the 4d $\mathcal{N}=2^*$ $U(N)$ gauge theory (a mass deformation of 4d $\mathcal{N}=4$ super-Yang-Mills theory), which is a generating series of equivariant integrals over instanton moduli spaces, is given by a sum over colored partitions weighted by a counting parameter $\mathfrak{q}$. This note proves convergence of the series in the unit disk $|\mathfrak{q}|<1$ for generic parameters. Specifically, the absolute convergence radius of this sum is determined, assuming that mass and Coulomb branch parameters avoid some lattice. If the ratio $b^2=\epsilon_1/\epsilon_2$ of equivariant parameters is in $\mathbb{C}\setminus[0,+\infty)$, the radius is $1$, as expected. If $b^2$ is non-negative, three cases arise: the radius is finite if $b^2$ has finite exponential type (a generalization of Brjuno numbers), namely there exists $C>0$ such that $|b^2-p/q|>\exp(-Cq)$ for all integers $p,q\neq 0$; the series diverges if $b^2$ is super-exponentially well approximable by rationals; and if $b^2$ is rational some terms are singular. The AGT correspondence translates these results to convergence of torus one-point conformal blocks of the Virasoro and $W_N$ algebras with non-real $b$, within the unit disk. For the Virasoro algebra this corresponds to a central charge in $\mathbb{C}\setminus[25,+\infty)$.