Extremal Correlators and Random Matrix Theory

TL;DR

This paper connects extremal correlators in 4D N=2 SCFTs to Wishart-Laguerre random matrix models, providing exact solutions and insights into large charge states, effective theories, and non-perturbative effects.

Contribution

It introduces a dual random matrix model approach to analyze extremal correlators in rank-one N=2 SCFTs, including exact solutions and large charge limit insights.

Findings

Large charge extremal correlators are described by Wishart-Laguerre random matrix models.

The model reveals a non-relativistic axion-dilaton effective theory in the large matrix limit.

Exact solutions are obtained for the first two orders in the 't Hooft expansion.

Abstract

We study the correlation functions of Coulomb branch operators of four-dimensional $\mathcal{N} = 2$ Superconformal Field Theories (SCFTs). We focus on rank-one theories, such as the SU(2) gauge theory with four fundamental hypermultiplets. "Extremal" correlation functions, involving exactly one anti-chiral operator, are perhaps the simplest nontrivial correlation functions in four-dimensional Quantum Field Theory. We show that the large charge limit of extremal correlators is captured by a "dual" description which is a chiral random matrix model of the Wishart-Laguerre type. This gives an analytic handle on the physics in some particular excited states. In the limit of large random matrices we find the physics of a non-relativistic axion-dilaton effective theory. The random matrix model also admits a 't Hooft expansion in which the matrix is taken to be large and simultaneously the coupling is taken to zero. This explains why the extremal correlators of SU(2) gauge theory obey a nontrivial double scaling limit in states of large charge. We give an exact solution for the first two orders in the 't Hooft expansion of the random matrix model and compare with expectations from effective field theory, previous weak coupling results, and we analyze the non-perturbative terms in the strong 't Hooft coupling limit. Finally, we apply the random matrix theory techniques to study extremal correlators in rank-1 Argyres-Douglas theories. We compare our results with effective field theory and with some available numerical bootstrap bounds.