Solving Virasoro constraints on integrable hierarchies via the Kontsevich-Miwa transform

Abstract

We solve Virasoro constraints on the KP hierarchy in terms of minimal conformal models. The constraints we start with are implemented by the Virasoro generators depending on a background charge Q. Then the solutions to the constraints are given by the theory which has the same field content as the David-Distler-Kawai theory: it consists of a minimal matter scalar with background charge Q, dressed with an extra “Liouville” scalar. In particular, the Virasoro-constrained τ-function is related to the correlator of a product of (dressed) “21” operators. The construction is based on a generalization of the Kontsevich parametrization of the KP times achieved by introducing into it Miwa parameters which depend on the value of Q. Under the thus defined Kontsevich-Miwa transformation, the Virasoro constraints are proven to be equivalent to a master equation depending on the parameter Q. The master equation is further identified with the decoupling equation corresponding to a null vector at level 2. We conjecture that higher-level decoupling equations are similarly related to constraints on the KP τ-function. We also consider the master equation for the N-reduced KP hierarchies. Several comments are made on a possible relation between the generalized master equation and scaled Kontsevich-type matrix integrals, and on the form the equation takes in higher genera.