Fusion of symmetric $D$-branes and Verlinde rings

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.

43 pages, xy-pic diagrams; Proof of Prop. 6.17 clarified and references added

References in corpus (5)

• Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories
• Twisted K-theory and loop group representations
• Abelian and non-Abelian branes in WZW models and gerbes
• D-branes in N=2 coset models and twisted equivariant K-theory
• K-theoretical boundary rings in N=2 coset models

Cited by in corpus (12)

• Can D-Branes Wrap Nonrepresentable Cycles?
• Higher U(1)-gerbe connections in geometric prequantization
• Dirac operators on quasi-Hamiltonian G-spaces
• Differential Twisted K-theory and Applications
• Dirac Geometry of the Holonomy Fibration
• Thom isomorphism and Push-forward map in twisted K-theory
• The ring structure for equivariant twisted K-theory
• Multiplicative Bundle Gerbes with Connection
• Twisted Conjugation on Connected Simple Lie Groups and Twining Characters
• Twisted K-homology and group-valued moment maps
• Classification of Module Categories for $SO(3{)}_{2m}$
• Verlinde modules and quantization