Integrable structures in matrix models and physics of 2d-gravity

TL;DR

This paper reviews how integrable structures like Virasoro-W constraints emerge in matrix models describing 2D gravity, highlighting their solutions, properties, and connections to continuum theories.

Contribution

It demonstrates the integrability of matrix models related to 2D gravity using determinant techniques and explores new solutions beyond generalized Kontsevich models.

Findings

Matrix models encode 2D gravity via Virasoro-W constraints.

Integrability is proven through determinant and free field techniques.

New insights into solutions and their relation to 2D gravity are presented.

Abstract

A review of the appearence of integrable structures in the matrix model description of $2d$-gravity is presented. Most of ideas are demonstrated at the technically simple but ideologically important examples. Matrix models are considered as a sort of "effective" description of continuum $2d$ field theory formulation. The main physical role in such description is played by the Virasoro-$W$ constraints which can be interpreted as a certain unitarity or factorization constraints. Bith discrete and continuum (Generalized Kontsevich) models are formulated as the solutions to those discrete (continuous) Virasoro-$W$ constraints. Their integrability properties are proven using mostly the determinant technique highly related to the representation in terms of free fields. The paper also contains some new observations connected to formulation of more general than GKM solutions and deeper understanding of their relation to $2d$ gravity.