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.

Higher U(1)-gerbe connections in geometric prequantization

“…It is a classical fact that a flat U (1)-bundle over a manifold X is equivalently the datum of a U (1)local system on X, i.e., of a U (1)-valued representation of the Poincaré groupoid of X. Translated into the language of smooth stacks, this becomes an instance of the ♭/Π-adjunction: 14 the space H(X, ♭BU (1)) of flat U (1)-connections on X is equivalently the space H(ΠX, BU (1)) of maps from the Poincaré groupoid of X to BU (1). Since BU (1) is a 1-stack, we have H(ΠX, BU (1)) ∼ = H((ΠX) ≤1 , BU (1)) see Section 2.9.…”

Multiplicative structures and the twisted Baum-Connes assembly map

“…Finally, the product (1.6) above is the first step into trying to understand, in the non proper case, internal stringy products in groups of the form K geo G, * (N, α) (or more generally on the K-theory counterpart) where N is a crossed module (for instance G itself on which G acts by conjugation, in the case of a group) over G and α a twisting with good multiplicative properties (transgressive). Indeed, in all the versions of stringy products (or internal products) one passes necessarily by a product as above before making use of the crossed module structure and of the multiplicativity of the twisting, [12], [2], [7], [23] for mention some of them.…”

The ring structure for equivariant twisted K-theory

“…al. [14] using bundle gerbes and was also used in [16] in their study of fusion of D-branes. The transgression map for orbifold cohomology was recently studied by AdemRuan-Zhang [1].…”

“…It would be interesting to explore the connection between the ring structure on K * [c],G (G) using our construction and the ones in [26] and [16].…”

“…is also standard [25,28,16]. The ring structures on twisted K-theory of orbifolds have been studied independently by Adem-Ruan-Zhang using a different method [1].…”