Affine Toda Field Theory in the Presence of Reflecting Boundaries

TL;DR

This paper derives explicit formulas for boundary scattering matrices in affine Toda field theories with reflecting boundaries, revealing two distinct solutions and their physical implications.

Contribution

It demonstrates that the boundary crossing-unitarity equation follows from bootstrap equations and provides explicit solutions for all simply laced affine Toda theories.

Findings

Explicit W-matrix formulas for all simply laced affine Toda theories

Identification of two distinct boundary scattering solutions per theory

Clarification of CDD-ambiguities in boundary conditions

Abstract

We show that the ``boundary crossing-unitarity equation" recently proposed by Ghoshal and Zamolodchikov is a consequence of the boundary bootstrap equation for the S-matrix and the wall-bootstrap equation. We solve this set of equations for all affine Toda theories related to simply laced Lie algebras, obtaining explicit formulas for the W-matrix which encodes the scattering of a particle with the boundary in the ground state. For each theory there are two solutions to these equations, related by CDD-ambiguities, each giving rise to different kind of physics.