Complex eigenvalue instantons and the Fredholm determinant expansion in the Gross-Witten-Wadia model

TL;DR

This paper analyzes nonperturbative effects in the Gross-Witten-Wadia model, identifying complex eigenvalue instantons as key contributors and relating them to Fredholm determinant expansions, with implications for supersymmetric gauge theories.

Contribution

It provides a detailed computation of leading nonperturbative corrections using eigenvalue tunneling and connects these to Fredholm determinant expansions in the GWW model.

Findings

Complex eigenvalue instantons are the dominant nonperturbative effects.

Fredholm determinant expansion matches the leading nonperturbative correction after the 't Hooft limit.

Ghost instantons are extrema on the unphysical sheet of the spectral curve.

Abstract

We study the leading nonperturbative corrections to the strong-coupling (ungapped) phase of the Gross-Witten-Wadia (GWW) integral over unitary matrices, to one-loop order. We compute these corrections directly in terms of eigenvalue tunneling in a holomorphic presentation of the integral over eigenvalues. The leading nonperturbative contribution to the partition function comes from a pair of complex eigenvalue instantons. We show that these are in fact "ghost instantons", which are extrema of the one-eigenvalue effective potential on the "unphysical sheet" of the spectral curve and have been discussed in detail recently by Mari\~no, Schiappa, and Schwick. Further, we discuss the relationship of these instantons to the Fredholm determinant expansion of the unitary matrix integral, which has recently become an object of interest in the computations of BPS indices of supersymmetric gauge theories and black holes. We find that, after taking the 't Hooft limit, the first correction given by the Fredholm determinant expansion of the GWW integral agrees precisely with the leading nonperturbative correction, to one-loop order.