Abelian and non-Abelian branes in WZW models and gerbes

TL;DR

This paper uses gerbes to classify and analyze boundary conditions and branes in WZW models, including their gauge fields and symmetry properties, with explicit formulas for boundary functions.

Contribution

It provides a geometric framework for classifying symmetric branes in WZW models with non-simply connected groups, including Abelian and non-Abelian gauge fields.

Findings

Classified branes supported by conjugacy classes with specific fundamental groups.

Derived explicit formulas for boundary partition functions.

Identified conditions for branes with twisted gauge fields.

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 Z_2\times Z_2 in SO(4n)/Z_2. 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.