Classically integrable boundary conditions for symmetric-space sigma 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 mode

Introduction to Yangian Symmetry in Integrable Field Theory

“…Boundary conditions for nonlinear sigma models in general seem to have been relatively little studied -see [111] and references therein. The more general theory of coideal subalgebras may be found in [112].…”

Algebraic Bethe Ansatz for O(2N) sigma models with integrable diagonal boundaries

“…Sigma models with integrability preserving boundaries were considered both in the classical field theoretic framework [4][5][6][7], and in the quantum theory [8][9][10], where the various reflection matrices were obtained. We take the diagonal ones and construct with them the double row monodromy and transfer matrices.…”

Boundary integrability of nonlinear sigma models

“…The key point, proved in [9], is that this must commute with σ for compatibility with the Poisson brackets and for conservation of the local charges. Then the gluing condition for the currents, analogous to (6) and (7), is…”

“…The details are given in [9]. The point is that with σ being the transposition operator in G × G, the only admissible nontrivial τ are (α, α) and σ(α, α) (with α being a freely chosen involution), and they respectively yield our two types of boundary condition: (2), (6) and (3), (7).…”

“…Here, we report work that began with the principal chiral field (on a compact Lie group G) [7], [8] and was then generalized to compact symmetric spaces G/H (with the principal chiral field realized as the symmetric space G × G/G) [9].…”