Large $N$ expansion of Wilson loops in the Gross-Witten-Wadia matrix model

TL;DR

This paper develops analytical methods to compute large N expansions of Wilson loops in the Gross-Witten-Wadia matrix model, including perturbative and instanton corrections, with explicit examples and general formulas.

Contribution

It introduces new analytical algorithms for precise calculation of Wilson loop corrections in the GWW model, extending to higher genus and instanton effects.

Findings

Genus five expansion of the one-cut resolvent captures all winding loops.

Periwal-Shevitz recursion generates all higher genus corrections.

Explicit formulas for next-to-leading corrections at general winding.

Abstract

We study the large $N$ expansion of winding Wilson loops in the off-critical regime of the Gross-Witten-Wadia (GWW) unitary matrix model. These have been recently considered in arXiv:1705.06542 and computed by numerical methods. We present various analytical algorithms for the precise computation of both the perturbative and instanton corrections to the Wilson loops. In the gapped phase of the GWW model we present the genus five expansion of the one-cut resolvent that captures all winding loops. Then, as a complementary tool, we apply the Periwal-Shevitz orthogonal polynomial recursion to the GWW model coupled to suitable sources and show how it generates all higher genus corrections to any specific loop with given winding. The method is extended to the treatment of instanton effects including higher order $1/N$ corrections. Several explicit examples are fully worked out and a general formula for the next-to-leading correction at general winding is provided. For the simplest cases, our calculation checks exact results from the Schwinger-Dyson equations, but the presented tools have a wider range of applicability.