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

TL;DR

This paper introduces bi-branes as submanifolds in product target spaces that describe topological defects in world sheet theories, linking geometric data with defect fusion and algebraic structures.

Contribution

It defines bi-branes as geometric objects for topological defects, explains their role in Wess-Zumino terms, and connects their fusion to algebraic structures like the Verlinde algebra.

Findings

Bi-branes are submanifolds called biconjugacy classes in WZW models.

The algebra of functions on biconjugacy classes relates to defect partition functions.

Fusion of bi-branes encodes the Verlinde algebra in WZW theories.

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.