Conformal Matrix Models as an Alternative to Conventional Multi-Matrix Models

TL;DR

This paper introduces conformal multi-matrix models as an alternative framework for 2D gravity with matter, demonstrating their equivalence to traditional models in the continuum limit and their solutions to integrable hierarchies.

Contribution

It proposes conformal multi-matrix models as a novel approach, solving extended Virasoro constraints and connecting to known 2D gravity models in the continuum limit.

Findings

CMM satisfy discrete W-constraints

CMM solutions correspond to (p,q)-series of 2D gravity

CMM generalize Hermitian one-matrix models

Abstract

We introduce {\it conformal multi-matrix models} (CMM) as an alternative to conventional multi-matrix model description of two-dimensional gravity interacting with $c < 1$ matter. We define CMM as solutions to (discrete) extended Virasoro constraints. We argue that the so defined alternatives of multi-matrix models represent the same universality classes in continuum limit, while at the discrete level they provide explicit solutions to the multi-component KP hierarchy and by definition satisfy the discrete $W$-constraints. We prove that discrete CMM coincide with the $(p,q)$-series of 2d gravity models in a {\it well}-{\it defined} continuum limit, thus demonstrating that they provide a proper generalization of Hermitian one-matrix model.