Integrable Boundary Conditions for the O(N) Nonlinear Sigma Model

Abstract

We discuss the new integrable boundary conditions for the O(N ) nonlinear σ model and related solutions of the boundary Yang-Baxter equation, which were presented in our previous paper hep-th/0108039.

Boundary integrability of nonlinear sigma models

“…Thus, for the CP N −1 or S N −1 model, we find that the corresponding CP M −1 or S M −1 (see[5]) is an admissible (nontrivial) D-submanifold for all M = 2, . For example, for the Grassmannian cases SU(N )/S(U (N − M ) × SU (M )), SO(N )/SO(N − M ) × SO(M ), or Sp(2n)/Sp(2n − 2m) × Sp(2m)), there exists a corresponding SU (N − M ) × SU (M ), SO,or Sp for any other M whose defining involution commutes with σ.…”

“…As previously stated, the present belief is that the bulk models are nonintegrable for nonsimple H, although recent results have reopened this question [24]; it seems likely that at least some such cases are quantum-integrable. The boundary quantum models, where they exist, have Y (g, h τ ) symmetry and can therefore use known solutions of the reflection equation, but the first priority is to establish the circumstances in which the quantum model is integrable, perhaps by extending the boundary version [5] of the anomaly-counting technique for the local charges [18] to general G/H. For the models with boundaries in symmetric spaces, at least, we now have a general statement about their classical integrability.…”

Boundary integrability of nonlinear sigma models

“…For the models with boundaries in symmetric spaces, at least, we now have a general statement about their classical integrability. The boundary quantum models, where they exist, have Y (g, h τ ) symmetry and can therefore use known solutions of the reflection equation, but the first priority is to establish the circumstances in which the quantum model is integrable, perhaps by extending the boundary version [5] of the anomaly-counting technique for the local charges [18] to general G/H.…”

“…For example, for the Grassmannian cases SU(N )/S(U (N − M ) × SU (M )), SO(N )/SO(N − M ) × SO(M ), or Sp(2n)/Sp(2n − 2m) × Sp(2m)), there exists a corresponding SU (N − M ) × SU (M ), SO,or Sp for any other M whose defining involution commutes with σ. Thus, for the CP N −1 or S N −1 model, we find that the corresponding CP M −1 or S M −1 (see[5]) is an admissible (nontrivial) D-submanifold for all M = 2, . .…”

On integrable boundaries in the 2 dimensional O(N) σ-models

“…Using the boundary generalizations of the Goldschmidt-Witten argument, Moriconi and de Martino [8] indicated that free and fixed boundary conditions can be quantum integrable and even extended the result for a mixture of free and fixed boundary conditions (for the boundary value of the fundamental field). Later Moriconi analyzed systematically the boundary conditions of the O(N ) models [9,10]. He identified new types of integrable boundary conditions, which can be implemented by adding a quadratic boundary potential including the time derivative of the fundamental field to the Lagrangian.…”