Classically integrable boundary conditions for symmetric-space sigma models

TL;DR

This paper studies boundary conditions that preserve integrability in symmetric-space sigma models, identifying conditions linked to involutions and applying results to specific models like $S^2$ and principal chiral models.

Contribution

It characterizes classically integrable boundary conditions for symmetric-space sigma models using involutions, extending known results to new settings.

Findings

Boundary conditions correspond to involutions commuting with the symmetric space involution.

For $S^2$, boundary conditions are great circles.

Reproduces known results for principal chiral models.

Abstract

We investigate boundary conditions for the nonlinear sigma model on the compact symmetric space $G/H$, where $H \subset G$ is the subgroup fixed by an involution $\sigma$ of $G$. The Poisson brackets and the classical local conserved charges necessary for integrability are preserved by boundary conditions in correspondence with involutions which commute with $\sigma$. Applied to $SO(3)/SO(2)$, the nonlinear sigma model on $S^2$, these yield the great circles as boundary submanifolds. Applied to $G \times G/G$, they reproduce known results for the principal chiral model.