Integrability of generalised type II defects in affine Toda field theory

TL;DR

This paper investigates the integrability of generalized type II defects in affine Toda field theories, focusing on conserved quantities and defect Lax matrices, particularly for the Tzitzéica and D4 models.

Contribution

It demonstrates the likely necessity of momentum conservation for integrability and derives defect Lax matrices ensuring zero curvature in specific ATFT defects.

Findings

Momentum conservation is likely necessary for integrability.

Lax matrices for Tzitzéica and D4 defects are constructed.

Additional defect potentials for D4 are identified.

Abstract

The Liouville integrability of the generalised type II defects is investigated. Full integrability is not considered, only the existence of an infinite number of conserved quantities associated with a system containing a defect. For defects in affine Toda field theories (ATFTs) it is shown that momentum conservation is very likely to be a necessary condition for integrability. The defect Lax matrices which guarantee zero curvature, and so an infinite number of conserved quantities, are calculated for the momentum conserving Tzitz\'eica defect and the momentum conserving $D_4$ ATFT defect. Some additional calculations pertaining to the $D_4$ defect are also carried out to find a more complete set of defect potentials than has appeared previously.