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

TL;DR

This paper explores the realization of the Virasoro algebra action within soliton theory, connecting it to KP symmetries, tau-functions, and infinite-dimensional flag spaces, revealing new representations and deformations.

Contribution

It introduces a novel realization of Virasoro algebra actions in soliton theory using Cauchy-Baker-Akhiezer kernels and constructs new tau-function representations on flag spaces.

Findings

Virasoro algebra representations with central charge 6j^2-6j+1 are generated by tau-function deformations.

The action of circle vector fields is realized through non-isospectral KP symmetries.

A combined KP and Toda lattice system with explicit Virasoro representations depending on an extra discrete time t_0.

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 \tau-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.