Solving Virasoro Constraints on Integrable Hierarchies via the Kontsevich-Miwa Transform

TL;DR

This paper solves Virasoro constraints on the KP hierarchy using a generalized Kontsevich-Miwa transform, linking them to a master equation and null-vector decoupling equations, with implications for matrix models and higher genus theories.

Contribution

It introduces a generalized Kontsevich-Miwa transformation to solve Virasoro constraints on integrable hierarchies, connecting them to decoupling equations and matrix models.

Findings

Virasoro constraints are equivalent to a master equation depending on background charge Q.

The master equation is identified with a null-vector decoupling equation.

Conjecture on W^{(n)} constraints relating to level-n decoupling equations.

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. 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 a null-vector decoupling equation. We conjecture that $W^{(n)}$ constraints on the KP hierarchy are similarly related to a level-$n$ decoupling equation. We also consider the master equation for the $N$-reduced KP hierarchies. Several comments are made on a possible relation of the generalized master equation to {\it scaled} Kontsevich-type matrix integrals and on the form the equation takes in higher genera.