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

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.