SDiff(2) Toda equation -- hierarchy, $\tau$ function, and symmetries

TL;DR

This paper introduces the SDiff(2) Toda equation as a continuum limit of the Toda lattice, establishing its Lax formalism, hierarchy, tau function, and symmetries related to area-preserving diffeomorphisms.

Contribution

It develops the Lax formalism, hierarchy, and tau function for the SDiff(2) Toda equation, revealing hidden symmetries and their algebraic structure, extending Toda theory.

Findings

Lax formalism and hierarchy established for SDiff(2) Toda equation

Tau function analogue introduced with symmetry properties

Hidden SDiff(2) symmetries derived from Riemann-Hilbert problem

Abstract

A continuum limit of the Toda lattice field theory, called the SDiff(2) Toda equation, is shown to have a Lax formalism and an infinite hierarchy of higher flows. The Lax formalism is very similar to the case of the self-dual vacuum Einstein equation and its hyper-K\"ahler version, however now based upon a symplectic structure and the group SDiff(2) of area preserving diffeomorphisms on a cylinder $S^1 \times \R$. An analogue of the Toda lattice tau function is introduced. The existence of hidden SDiff(2) symmetries are derived from a Riemann-Hilbert problem in the SDiff(2) group. Symmetries of the tau function turn out to have commutator anomalies, hence give a representation of a central extension of the SDiff(2) algebra.