Non-Linear Sigma Models on a Half Plane

TL;DR

This paper investigates the classical integrability of various two-dimensional non-linear sigma models on a half plane, revealing that boundary conditions generally break integrability unless trivial solutions are considered.

Contribution

It demonstrates that non-local charges characterizing integrability are not conserved under typical boundary conditions, highlighting the impact of boundaries on these models' integrability.

Findings

Non-local charges are not conserved with free boundary conditions.

Conservation of non-local charges leads to trivial solutions.

Most non-linear sigma models lose classical integrability on a half plane.

Abstract

In the context of integrable field theory with boundary, the integrable non-linear sigma models in two dimensions, for example, the $O(N)$, the principal chiral, the ${\rm CP}^{N-1}$ and the complex Grassmannian sigma models are discussed on a half plane. In contrast to the well known cases of sine-Gordon, non-linear Schr\"odinger and affine Toda field theories, these non-linear sigma models in two dimensions are not classically integrable if restricted on a half plane. It is shown that the infinite set of non-local charges characterising the integrability on the whole plane is not conserved for the free (Neumann) boundary condition. If we require that these non-local charges to be conserved, then the solutions become trivial.