Quantum integrability in two-dimensional systems with boundary

TL;DR

This paper investigates the integrability of affine Toda systems on a half-plane, analyzing classical and quantum boundary effects, and identifies conditions for conserved currents with boundary perturbations.

Contribution

It formulates a Lax pair approach for classical integrability with boundaries and examines quantum corrections, highlighting the need for boundary potential renormalization.

Findings

Classical integrability is compatible with specific boundary perturbations.

Quantum corrections often require boundary potential renormalization.

The sinh-Gordon model uniquely maintains quantum conserved currents without renormalization.

Abstract

In this paper we consider affine Toda systems defined on the half-plane and study the issue of integrability, i.e. the construction of higher-spin conserved currents in the presence of a boundary perturbation. First at the classical level we formulate the problem within a Lax pair approach which allows to determine the general structure of the boundary perturbation compatible with integrability. Then we analyze the situation at the quantum level and compute corrections to the classical conservation laws in specific examples. We find that, except for the sinh-Gordon model, the existence of quantum conserved currents requires a finite renormalization of the boundary potential.