Flag Spaces in KP Theory and Virasoro Action on \det D_j and Segal-Wilson τ-Function

Abstract

It is well-known that the algebra of vector fields on the circle acts on the space of Riemann surfaces with a marked point and a local parameter at this point. We show that this action has a natural realization in the soliton theory, indeed it coincides with the action of some non-isospectral Kadomtsev-Petviashvili symmetries on the finite-gap solutions. A technique based on the so-called Cauchy-Baker-Akhiezer kernel is developed. The deformations of the τ-function corresponding to the Baker-Akhiezer forms of tensor weight j generate representations of the Virasoro algebra with a central charge 6j^2-6j+1. A system including the Kadomtsev-Petviashvili hierarchy and the Toda lattice simultaneously is considered. The Virasoro representations corresponding to such a system explicitly depend on an extra discrete time t_0. The tau-function for this system is defined in terms of infinite dimensional flag spaces, generalizing the grassmanians.