Affine Toda field theories with defects

TL;DR

This paper develops a Lagrangian framework for studying defects in affine Toda field theories, revealing that certain models preserve integrability and modified conserved quantities despite explicit symmetry breaking.

Contribution

It introduces a Lagrangian approach with a constructed Lax pair for affine Toda theories with defects, identifying models that maintain integrability and conserved charges.

Findings

Only models based on a_r^{(1)} data preserve integrability with defects.

Modified energy and momentum are conserved despite symmetry breaking.

Some higher spin charges are preserved with defect contributions.

Abstract

A Lagrangian approach is proposed and developed to study defects within affine Toda field theories. In particular, a suitable Lax pair is constructed together with examples of conserved charges. It is found that only those models based on $a_r^{(1)}$ data appear to allow defects preserving integrability. Surprisingly, despite the explicit breaking of Lorentz and translation invariance, modified forms of both energy and momentum are conserved. Some, but apparently not all, of the higher spin conserved charges are also preserved after the addition of contributions from the defect. This fact is illustrated by noting how defects may preserve a modified form of just one of the spin 2 or spin -2 charges but not both of them.