Integrable boundary conditions and reflection matrices for the O(N) nonlinear sigma model

Abstract

We find new integrable boundary conditions, depending on a free parameter g, for the O(N ) nonlinear σ model, which are of nondiagonal type, that is, particles can change their "flavor" through scattering off the boundary. These boundary conditions are derived from a microscopic boundary lagrangian, which is used to establish their integrability, and exhibit integrable flows between diagonal boundary conditions investigated previously. We solve the boundary Yang-Baxter equation, connect these solutions to the boundary conditions, and examine the corresponding integrable flows.1

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

“…In [2] we found other solutions to the bYBe for the O(2N ) nlσ model, but were not able to link them to any boundary conditions. Another special case is the O(2) nlσ model.…”

“…Once the ratios X(θ) and Y (θ) have been fixed, all that is left to do is to compute the overall factor for the reflection matrix, which can be done with the use of boundary unitarity and boundary crossingsymmetry, and a minimality hypothesis for the pole structure of the reflection matrix. We refer to [2] for the explicit results. Note that if c = c ′ the off-diagonal amplitudes ±Y (θ) vanish, and we recover a diagonal scattering problem.…”

“…In [2] we have found new integrable boundary conditions that break the bulk O(N ) symmetry to O(2) × O(N − 2) at the boundary, and which depend on one free-parameter g. The reason for this symmetry at the boundary is the following. Free boundary conditions (Neumann) have O(N ) boundary symmetry.…”

“…The following discussion is informal and intended to a general audience, we refer to [2] for a more complete discussion.…”

“…The equation of motion for the field n is then easily derived from (2) and, in light-cone coordinates x ± = (x 0 ± x 1 )/2, it reads…”