Correlation Functions in Unitary Minimal Liouville Gravity and Frobenius Manifolds

TL;DR

This paper explores correlation functions in unitary minimal Liouville gravity using Frobenius manifolds, confirming some theoretical predictions and highlighting unresolved issues regarding selection rules and correlator interpretations.

Contribution

It applies Frobenius manifold solutions to compute correlators in MLG, aligning with previous methods and revealing limitations of resonance transformations.

Findings

Agreement with original approach in applicable parameter regions

Partial satisfaction of conformal selection rules

Identification of nonzero correlators violating initial rules

Abstract

We continue to study minimal Liouville gravity (MLG) using a dual approach based on the idea that the MLG partition function is related to the tau function of the A_q integrable hierarchy via the resonance transformations, which are in turn fixed by conformal selection rules. One of the main problems in this approach is to choose the solution of the Douglas string equation that is relevant for MLG. The appropriate solution was recently found using connection with the Frobenius manifolds. We use this solution to investigate three- and four-point correlators in the unitary MLG models. We find an agreement with the results of the original approach in the region of the parameters where both methods are applicable. In addition, we find that only part of the selection rules can be satisfied using the resonance transformations. The physical meaning of the nonzero correlators, which before coupling to Liouville gravity are forbidden by the selection rules, and also the modification of the dual formulation that takes this effect into account remains to be found.