Generalized Toda Field Theories

TL;DR

This paper introduces a unified framework for Toda field theories parametrized by a continuous variable, interpolating between standard models and new integrable systems with infinite fields and higher-spin currents.

Contribution

It formulates a generalized class of Toda models with an arbitrary parameter, revealing new integrable systems with conformal invariance and infinite conserved currents.

Findings

Models include standard Toda theories for specific parameter values

New models involve infinitely many fields and are conformally invariant

Explicit Poisson structure for the currents is derived

Abstract

In this paper we introduce a unified approach to Toda field theories which allows us to formulate the classes of $A_n$, $B_n$ and $C_n$ models as unique models involving an arbitrary continuous parameter $\nu$. For certain values of $\nu $, the model describes the standard Toda theories. For other values of $\nu$ it defines a class of models that involve infinitely many fields. These models interpolate between the various standard Toda field theories. They are conformally invariant and possess infinitely many conserved higher-spin currents thus making them candidates for a new set of integrable systems. A general construction is performed, which can effectively be used for the derivation of explicit forms of particular higher-spin currents. We also study the currents in a different representation in which they are linear in the dynamical variables having, however, a non-linear Poisson bracket algebra. An explicit formula for this Poisson structure is found.