A Conformal Field Theory Formalism from Integrable Hierarchies via the Kontsevich--Miwa Transform

TL;DR

This paper derives a conformal field theory framework for 2D quantum gravity coupled with matter from integrable hierarchies using a generalized Kontsevich--Miwa transform, connecting it to the David-Distler-Kawai formalism.

Contribution

It introduces a novel approach to derive 2D quantum gravity and matter CFTs directly from integrable hierarchies via a generalized Kontsevich--Miwa transform, linking to established formalisms.

Findings

Recovered the matter field content with background charge Q_m

Related the tau function to correlators of minimal model operators

Established a connection between integrable hierarchies and 2D quantum gravity

Abstract

We attempt a direct derivation of a conformal field theory description of 2D quantum gravity~+~matter from the formalism of integrable hierarchies subjected to Virasoro constraints. The construction is based on a generalization of the Kontsevich parametrization of the KP times by introducing Miwa parameters into it. The resulting Kontsevich--Miwa transform can be applied to the Virasoro constraints provided the Miwa parameters are related to the background charge $Q$ of the Virasoro generators on the hierarchy. We then recover the field content of the David-Distler-Kawai formalism, with the matter theory represented by a scalar with the background charge $Q_m=Q-{Q\over 2}$. In particular, the tau function is related to the correlator of a product of the `21' operators of the minimal model with central charge $d=1-3Q_m^2$.