Some remarks on D-branes and defects in Liouville and Toda field theories

TL;DR

This paper investigates the structure of boundary states and defects in Liouville and Toda conformal field theories, demonstrating simplified equations and confirming recent conjectures about defects and boundary states across these models.

Contribution

It derives the Cardy-Lewellen equation for all $sl(n)$ Toda theories and confirms the validity of recently proposed boundary states in these models.

Findings

Cardy-Lewellen equation simplifies in diagonal models due to conformal properties.

Recent defect conjectures in Toda theories are proven to satisfy the cluster equation.

Boundary states in $sl(3)$ Toda theory are shown to extend to all $sl(n)$ theories.

Abstract

In this paper we analyze the Cardy-Lewellen equation in general diagonal model. We show that in these models it takes simple form due to some general properties of conformal field theories, like pentagon equations and OPE associativity. This implies, that the Cardy-Lewellen equation has simple form also in non-rational diagonal models. We specialize our finding to the Liouville and Toda field theories. In particular we prove, that conjectured recently defects in Toda field theory indeed satisfy the cluster equation. We also derive the Cardy-Lewellen equation in all $sl(n)$ Toda field theories and prove that the forms of boundary states found recently in $sl(3)$ Toda field theory hold in all $sl(n)$ theories as well.