Complex (super)-matrix models with external sources and $q$-ensembles of Chern-Simons and ABJ(M) type

TL;DR

This paper establishes a connection between complex matrix models with external sources and $q$-ensembles, showing how they compute probabilities of large fluctuations in Chern-Simons and ABJ(M) theories, extending to supermatrix models.

Contribution

It demonstrates that the LSZ matrix model and its supermatrix extension describe large fluctuation probabilities in Chern-Simons and ABJ(M) matrix models, linking complex matrix models with topological quantum field theories.

Findings

LSZ matrix model computes large fluctuation probabilities in $q$-ensembles.

Extension of results to supermatrix models for ABJ(M) theories.

Connection between external source spectra and topological invariants.

Abstract

The Langmann-Szabo-Zarembo (LSZ) matrix model is a complex matrix model with a quartic interaction and two external matrices. The model appears in the study of a scalar field theory on the non-commutative plane. We prove that the LSZ matrix model computes the probability of atypically large fluctuations in the Stieltjes-Wigert matrix model, which is a $q$-ensemble describing $U(N)$ Chern-Simons theory on the three-sphere. The correspondence holds in a generalized sense: depending on the spectra of the two external matrices, the LSZ matrix model either describes probabilities of large fluctuations in the Chern-Simons partition function, in the unknot invariant or in the two-unknot invariant. We extend the result to supermatrix models, and show that a generalized LSZ supermatrix model describes the probability of atypically large fluctuations in the ABJ(M) matrix model.