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

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 $Q_m=Q-\frac{Q}{2}$. In particular, the tau function is related to the correlator of a product of the `21' operators of the minimal model with central charge $d=1-3Q_m^2$.

10 pages

References in corpus (2)

• On Equivalence of Topological and Quantum 2d Gravity
• Kontsevich - Miwa Transform of the Virasoro Constraints as Null-Vector Decoupling Equations