Boundary Lax pairs for the $A_{n}^{(1)}$ Toda field theories

TL;DR

This paper develops a systematic method to construct boundary Lax pairs for $A_n^{(1)}$ affine Toda field theories, explicitly working out the sine-Gordon and $A_2^{(1)}$ models, including novel boundary conditions.

Contribution

It introduces a general scheme for boundary Lax pairs in $A_n^{(1)}$ ATFTs and explicitly constructs models with new boundary conditions, especially for the $A_2^{(1)}$ case.

Findings

Explicit boundary Lax pairs for sine-Gordon and $A_2^{(1)}$ models.

First examples of boundary conditions with two distinct types.

Novel expressions for boundary Lax pairs.

Abstract

Based on the recent formulation of a general scheme to construct boundary Lax pairs,we develop this systematic construction for the $A_n^{(1)}$ affine Toda field theories (ATFT). We work out explicitly the first two models of the hierarchy, i.e. the sine-Gordon ($A_1^{(1)}$) and the $A_2^{(1)}$ models. The $A_2^{(1)}$ Toda theory is the first non-trivial example of the hierarchy that exhibits two distinct types of boundary conditions. We provide here novel expressions of boundary Lax pairs associated to both types of boundary conditions.