The Lie-Poisson Structure of Integrable Classical Non-Linear Sigma Models

TL;DR

This paper reveals that classical non-linear sigma models on symmetric spaces have a Poisson structure described by an $r$-$s$-matrix formalism, introducing a new algebraic framework that generalizes the classical Yang-Baxter algebra for integrable models.

Contribution

It explicitly computes the $r$ and $s$ matrices for these models and proposes a new algebraic structure replacing the classical Yang-Baxter algebra for non-ultralocal integrable systems.

Findings

Explicit $r$ and $s$ matrices are derived.

A new algebraic structure is proposed for non-ultralocal models.

Poisson brackets for transition matrices are calculated.

Abstract

The canonical structure of classical non-linear sigma models on Riemannian symmetric spaces, which constitute the most general class of classical non-linear sigma models known to be integrable, is shown to be governed by a fundamental Poisson bracket relation that fits into the $r$-$s$-matrix formalism for non-ultralocal integrable models first discussed by Maillet. The matrices $r$ and $s$ are computed explicitly and, being field dependent, satisfy fundamental Poisson bracket relations of their own, which can be expressed in terms of a new numerical matrix~$c$. It is proposed that all these Poisson brackets taken together are representation conditions for a new kind of algebra which, for this class of models, replaces the classical Yang-Baxter algebra governing the canonical structure of ultralocal models. The Poisson brackets for the transition matrices are also computed, and the notorious regularization problem associated with the definition of the Poisson brackets for the monodromy matrices is discussed.