Continuum limit of the Volterra model, separation of variables and non standard realizations of the Virasoro Poisson bracket

TL;DR

This paper explores the continuum limit of the integrable Volterra model, revealing how it relates to the Virasoro algebra through determinants of infinite matrices and separated variables on Riemann surfaces, with explicit solutions for eigenvectors.

Contribution

It introduces a novel continuum limit of the Volterra model that yields non-standard realizations of the Virasoro Poisson bracket using infinite matrices and Riemann surface points.

Findings

Virasoro generators expressed as determinants of infinite matrices

Explicit calculation of the continuum limit of eigenvectors

Schrödinger equation associated with the model is exactly solvable

Abstract

The classical Volterra model, equipped with the Faddeev-Takhtadjan Poisson bracket provides a lattice version of the Virasoro algebra. The Volterra model being integrable, we can express the dynamical variables in terms of the so called separated variables. Taking the continuum limit of these formulae, we obtain the Virasoro generators written as determinants of infinite matrices, the elements of which are constructed with a set of points lying on an infinite genus Riemann surface. The coordinates of these points are separated variables for an infinite set of Poisson commuting quantities including $L\_0$. The scaling limit of the eigenvector can also be calculated explicitly, so that the associated Schroedinger equation is in fact exactly solvable.