The geometry of WZW branes

TL;DR

This paper explores the geometric structure of WZW branes, revealing how boundary conditions relate to conjugacy classes and algebraic properties of open strings, with implications for conformal field theory.

Contribution

It provides a detailed analysis of the target space geometry of WZW branes and links algebraic structures to geometric configurations at finite and large levels.

Findings

Branes correspond to conjugacy classes in the large level limit.

Open strings connecting different branes vanish at large level.

Boundary operator algebra constants relate to fusing matrices in rational CFTs.

Abstract

The structures in target space geometry that correspond to conformally invariant boundary conditions in WZW theories are determined both by studying the scattering of closed string states and by investigating the algebra of open string vertex operators. In the limit of large level, we find branes whose world volume is a regular conjugacy class or, in the case of symmetry breaking boundary conditions, a `twined' version thereof. In particular, in this limit one recovers the commutative algebra of functions over the brane world volume, and open strings connecting different branes disappear. At finite level, the branes get smeared out, yet their approximate localization at (twined) conjugacy classes can be detected unambiguously. As a by-product, it is demonstrated how the pentagon identity and tetrahedral symmetry imply that in any rational conformal field theory the structure constants of the algebra of boundary operators coincide with specific entries of fusing matrices.