SOME REMARKS ON D-BRANES AND DEFECTS IN LIOUVILLE AND TODA FIELD THEORIES

In this paper, we analyze the Cardy-Lewellen equation in general diagonal model. We show that in these models it takes a 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 a simple form also in nonrational diagonal models. We specialize our finding to the Liouville and Toda field theories. In particular, we prove that recently conjectured 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 form of boundary states found recently in sl(3) Toda field theory holds in all sl(n) theories as well.