Fusion of symmetric $D$-branes and Verlinde rings

TL;DR

This paper explores how multiplicative bundle gerbes over compact Lie groups induce a fusion category of equivariant modules, connecting D-branes, quasi-Hamiltonian spaces, and Verlinde rings, extending the Freed-Hopkins-Teleman theorem.

Contribution

It introduces a new fusion category framework linking bundle gerbes, D-branes, and Verlinde rings, especially for non-simply connected groups.

Findings

Established a fusion category from equivariant bundle gerbe modules.

Connected D-brane fusion with Verlinde ring structures.

Extended analysis to non-simply connected Lie groups.

Abstract

We explain how multiplicative bundle gerbes over a compact, connected and simple Lie group $G$ lead to a certain fusion category of equivariant bundle gerbe modules given by pre-quantizable Hamiltonian $LG$-manifolds arising from Alekseev-Malkin-Meinrenken's quasi-Hamiltonian $G$-spaces. The motivation comes from string theory namely, by generalising the notion of $D$-branes in $G$ to allow subsets of $G$ that are the image of a $G$-valued moment map we can define a `fusion of $D$-branes' and a map to the Verlinde ring of the loop group of $G$ which preserves the product structure. The idea is suggested by the theorem of Freed-Hopkins-Teleman. The case where $G$ is not simply connected is studied carefully in terms of equivariant bundle gerbe modules for multiplicative bundle gerbes.