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

A. M. Semikhatov
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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 $d=1-3Q_m^2$.

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A. M. Semikhatov. 1991-12-19. A Conformal Field Theory Formalism from Integrable Hierarchies via the Kontsevich--Miwa Transform. https://arxiv.org/abs/hep-th/9112055

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