Frobenius manifolds, Integrable Hierarchies and Minimal Liouville Gravity

TL;DR

This paper explores the relationship between Frobenius manifolds and Minimal Liouville gravity, providing explicit solutions to the Douglas string equation that satisfy conformal and fusion rules in flat coordinates.

Contribution

It introduces a new explicit solution to the Douglas string equation in Frobenius manifold coordinates for (p,q) Minimal Liouville gravity, linking integrable hierarchies with quantum gravity models.

Findings

Explicit simple form of the solution in flat coordinates

Successful transformation from KdV to Liouville frame

Fulfillment of conformal and fusion rules

Abstract

We use the connection between the Frobrenius manifold and the Douglas string equation to further investigate Minimal Liouville gravity. We search a solution of the Douglas string equation and simultaneously a proper transformation from the KdV to the Liouville frame which ensure the fulfilment of the conformal and fusion selection rules. We find that the desired solution of the string equation has explicit and simple form in the flat coordinates on the Frobenious manifold in the general case of (p,q) Minimal Liouville gravity.