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

TL;DR

The article introduces conformal multi-matrix models (CMM) as discrete, conformal realizations of $c<1$ 2d gravity that satisfy extended Virasoro/$W$ constraints, arguing these models belong to the same continuum universality class as conventional multi-matrix models while exposing a richer integrable structure via multi-component KP (AKNS-type) reductions. It provides a determinant (tau-function) representation and a free-fermion construction, demonstrating that CMM are tau-functions of a multi-component KP hierarchy constrained by discrete $W$-algebras, and it develops a systematic double-scaling procedure that maps discrete $W$-constraints to their continuum counterparts through Kazakov-type time transformations and basis changes in the Cartan plane. The paper works out explicit $p=3$ examples (two-matrix case) and generalizes to arbitrary $p$, clarifying how reductions yield continuum Virasoro and $W$-algebras, and discusses contour prescriptions and the relation to Generalized Kontsevich Models. These results illuminate the integrable origin of 2d gravity in a discrete conformal setting and point toward a unified, GKM-amenable framework for studying continuum limits. The work suggests practical computational avenues for accessing continuum $W$-invariants and highlights open questions about the precise conformal-field interpretation of the inserted screenings.

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.