Bi-branes: Target Space Geometry for World Sheet topological Defects

TL;DR

This work identifies world-sheet topological defects with bi-branes, submanifolds of a product target space $M_1{\times}M_2$ equipped with (twisted) bundles, and develops a geometric framework for their Wess-Zumino terms and fusion. In WZW theories, defects correspond to biconjugacy classes, whose world-volume quantization produces a $G\times G$-module structure that matches defect-field algebras and, in the large-level limit, yields Verlinde-type fusion data. The paper provides both a concrete target-space description for defects in current-algebra CFTs and a general, cohomological construction of WZ terms via bundle gerbes and bimodules, along with explicit fusion rules for simple models such as the compactified free boson and WZW models. The results bridge conformal defect data with geometric quantization of bi-branes, offering a path to generalized dualities and potential connections to broader structures like Langlands duality.

Abstract

We establish that the relevant geometric data for the target space description of world sheet topological defects are submanifolds - which we call bi-branes - in the product M1 x M2 of the two target spaces involved. Very much like branes, they are equipped with a vector bundle, which in backgrounds with non-trivial B-field is actually a twisted vector bundle. We explain how to define Wess-Zumino terms in the presence of bi-branes and discuss the fusion of bi-branes. In the case of WZW theories, symmetry preserving bi-branes are shown to be biconjugacy classes. The algebra of functions on a biconjugacy class is shown to be related, in the limit of large level, to the partition function for defect fields. We finally indicate how the Verlinde algebra arises in the fusion of WZW bi-branes.