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arXiv · hep-th/9204063

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

A. M. Semikhatov
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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.

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A. M. Semikhatov. 1992-04-20. Solving Virasoro Constraints on Integrable Hierarchies via the Kontsevich-Miwa Transform. https://doi.org/10.1016/0550-3213(92)90178-e

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