Reflection Bootstrap Equations for Toda Field Theory

TL;DR

This paper develops algebraic equations called Reflection Bootstrap Equations for Toda field theories with boundaries, providing explicit reflection matrices and symmetry properties for various models, advancing understanding of integrable boundary quantum field theories.

Contribution

It introduces Reflection Bootstrap Equations specific to Toda field theories, offering explicit solutions and symmetry analysis for boundary integrable models, extending prior algebraic frameworks.

Findings

Derived explicit reflection matrices for various Toda theories.

Analyzed symmetry properties of reflection matrices.

Connected Reflection Bootstrap Equations to known boundary conditions.

Abstract

An algebraic approach to integrable quantum field theory with a boundary (a half line) is presented and interesting algebraic equations, Reflection equations (RE) and Reflection Bootstrap equations (RBE) are discussed. The Reflection equations are a consistent generalisation of Yang-Baxter equations for factorisable scatterings on a half line (or with a reflecting boundary). They determine the so-called reflection matrices. However, for Toda field theory and/or other theories with diagonal S-matrices, the Reflection-Bootstrap equations proposed by Fring and K\"oberle determine the reflection matrices, since the reflection equations and the Yang-Baxter equations become trivial in these cases. The explicit forms of the reflection matrices together with their symmetry properties are given for various Toda field theories, simply laced and non-simply laced.