Abelian and Non-Abelian Branes in WZW Models and Gerbes

Abstract

We discuss how gerbes may be used to set up a consistent Lagrangian approach to the WZW models with boundary. The approach permits to study in detail possible boundary conditions that restrict the values of the fields on the worldsheet boundary to brane submanifolds in the target group. Such submanifolds are equipped with an additional geometric structure that is summarized in the notion of a gerbe module and includes a twisted Chan-Paton gauge field. Using the geometric approach, we present a complete classification of the branes that conserve the diagonal current-algebra symmetry in the WZW models with simple, compact but not necessarily simply connected target groups. Such symmetric branes are supported by a discrete series of conjugacy classes in the target group and may carry Abelian or non-Abelian twisted gauge fields. The latter situation occurs for the conjugacy classes with fundamental group ℤ2×ℤ2 in SO(4n)/ℤ2. The branes supported by such conjugacy classes have to be equipped with a projectively flat twisted U(2) gauge field in one of the two possible WZW models differing by discrete torsion. We show how the geometric description of branes leads to explicit formulae for the boundary partition functions and boundary operator product coefficients in the WZW models with non-simply connected target groups.