Unitary Minimal Liouville Gravity and Frobenius Manifolds

TL;DR

This paper explores the coupling of unitary minimal models to Liouville gravity through the Douglas string equation, utilizing Frobenius manifold structures to compute correlation functions and resonance transformations.

Contribution

It introduces a novel approach using Frobenius manifold flat coordinates to solve the Douglas string equation for minimal Liouville gravity.

Findings

Identified the appropriate solution of the Douglas string equation in flat coordinates.

Derived structure constants of the Frobenius algebra in canonical and flat coordinates.

Expressed resonance transformation coefficients using Jacobi polynomials.

Abstract

We study unitary minimal models coupled to Liouville gravity using Douglas string equation. Our approach is based on the assumption that there exist an appropriate solution of the Douglas string equation and some special choice of the resonance transformation such that necessary selection rules of the minimal Liouville gravity are satisfied. We use the connection with the Frobenius manifold structure. We argue that the flat coordinates on the Frobenius manifold are the most appropriate choice for calculating correlation functions. We find the appropriate solution of the Douglas string equation and show that it has simple form in the flat coordinates. Important information is encoded in the structure constants of the Frobenius algebra. We derive these structure constants in the canonical coordinates and in the physically relevant domain in the flat coordinates. We find the leading terms of the resonance transformation and express the coefficients of the resonance transformation in terms of Jacobi polynomials.