Ward Identities of $W_{\infty}$ Symmetry in Liouville Theory coupled to $c_M < 1$ Matter

TL;DR

This paper explores the Ward identities associated with $W_{}$ symmetry in Liouville theory coupled with minimal matter, revealing their equivalence to $W_q$ algebra constraints derived from matrix models.

Contribution

It demonstrates the equivalence between Ward identities in Liouville theory with matter and $W_q$ algebra constraints from matrix models, using analytic continuation.

Findings

Ward identities match $W_q$ algebra constraints

Correlation functions are defined via analytic continuation

Establishes a link between Liouville theory and matrix models

Abstract

We investigate the Ward identities of the $W_{\infty}$ symmetry in the Liouville theory coupled to the $(p,q)$ conformal matter. The correlation functions are defined by applying the analytic continuation procedure for the matter sector as well as the Liouville one. We then find that the Ward identities are equivalent to the $W_q$ algebra constraints deduced from the matrix model.