Matrix model approach to minimal Liouville gravity revisited

TL;DR

This paper revisits the matrix model approach to minimal Liouville gravity, proposing modifications to the Frobenius manifold structure to better encode conformal selection rules and matching correlators with continuous methods.

Contribution

It introduces a modified Frobenius manifold construction for the Lee-Yang series, enabling exact discrete formulations that align with continuous results without resonance transformations.

Findings

Correlators on the sphere up to four points match continuous approach.

Modified $A_{p-1}$ algebra improves the matrix model description.

Exact discrete formulation captures conformal selection rules.

Abstract

Using the connection with the Frobenius manifold structure, we study the matrix model description of minimal Liouville gravity (MLG) based on the Douglas string equation. Our goal is to find an exact discrete formulation of the (q,p) MLG model that intrinsically contains information about the conformal selection rules. We discuss how to modify the Frobenius manifold structure appropriately for this purposes. We propose a modification of the construction for Lee-Yang series involving the $A_{p-1}$ algebra instead of the previously used $A_1$ algebra. With the new prescription, we calculate correlators on the sphere up to four points and find full agreement with the continuous approach without using resonance transformations.