Modular anomaly equations in N=2* theories and their large-N limit

TL;DR

This paper introduces a modular anomaly equation for the prepotential of N=2* super Yang-Mills theory, analyzes its large-N behavior with instantons, and derives exact formulas for the prepotential in specific vacua, revealing simplified instanton contributions.

Contribution

It presents a novel modular anomaly equation for the prepotential and explores its implications in the large-N limit with exact instanton formulas for specific vacua.

Findings

Large-N limit with fixed coupling shows instantons are not suppressed.

Exact all-instanton formulas derived for specific vacua distributions.

Instanton contributions become independent of coupling at large N.

Abstract

We propose a modular anomaly equation for the prepotential of the N=2* super Yang-Mills theory on R^4 with gauge group U(N) in the presence of an Omega-background. We then study the behaviour of the prepotential in a large-N limit, in which N goes to infinity with the gauge coupling constant kept fixed. In this regime instantons are not suppressed. We focus on two representative choices of gauge theory vacua, where the vacuum expectation values of the scalar fields are distributed either homogeneously or according to the Wigner semi-circle law. In both cases we derive an all-instanton exact formula for the prepotential. As an application, we show that the gauge theory partition function on S^4 at large N localises around a Wigner distribution for the vacuum expectation values leading to a very simple expression in which the instanton contribution becomes independent of the coupling constant.