Ward Identities of Liouville Gravity coupled to Minimal Conformal Matter

TL;DR

This paper investigates the Ward identities in Liouville gravity coupled with minimal conformal matter, introducing pseudo-null fields and generalized equations of motion, and demonstrates their consistency with matrix model results.

Contribution

It introduces pseudo-null fields and generalized equations of motion, linking Ward identities with W and Virasoro constraints in Liouville gravity.

Findings

Explicit evaluation of boundary contributions.

Realization of structures similar to W and Virasoro constraints.

Solutions consistent with matrix model results.

Abstract

The Ward identities of the Liouville gravity coupled to the minimal conformal matter are investigated. We introduce the pseudo-null fields and the generalized equations of motion, which are classified into series of the Liouville charges. These series have something to do with the W and Virasoro constraints. The pseudo-null fields have non-trivial contributions at the boundaries of the moduli space. We explicitly evaluate the several boundary contributions. Then the structures similar to the W and the Virasoro constraints appearing in the topological and the matrix methods are realized. Although our Ward identities have some different features from the other methods, the solutions of the identities are consistent to the matrix model results.