Solving four-dimensional superconformal Yang-Mills theories with Tracy-Widom distribution

TL;DR

This paper links special observables in superconformal Yang-Mills theories to Tracy-Widom distributions, enabling efficient calculations at various couplings and revealing resurgence structures in their expansions.

Contribution

It introduces a novel method connecting superconformal Yang-Mills observables to random matrix theory, specifically Tracy-Widom distributions, for both weak and strong coupling regimes.

Findings

Derived explicit expansion coefficients for strong coupling series.

Established the connection between observables and Tracy-Widom distribution.

Demonstrated resurgence relations linking perturbative and non-perturbative parts.

Abstract

We study a special class of observables in $\mathcal N=2$ and $\mathcal N=4$ superconformal Yang-Mills theories which, for an arbitrary 't Hooft coupling constant $\lambda$, admit representation as determinants of certain semi-infinite matrices. Similar determinants have previously appeared in the study of level-spacing distributions in random matrices and are closely related to the celebrated Tracy-Widom distribution. We exploit this relationship to develop an efficient method for computing the observables in superconformal Yang-Mills theories at both weak and strong coupling. The weak coupling expansion has a finite radius of convergence. The strong coupling expansion involves the sum of the `perturbative' part, given by series in $1/\sqrt\lambda$, and the `non-perturbative' part, given by an infinite sum of exponentially small terms, each accompanied by a series in $1/\sqrt\lambda$ with factorially growing coefficients. We explicitly compute the expansion coefficients of these series and show that they are uniquely determined by the large order behavior of the expansion coefficients of the perturbative part via resurgence relations.