Abstract
In this paper, we identify the Ad-equivariant twisted $K$-theory of a compact Lie group $G$ with the “Verlinde group” of isomorphism classes of admissible representations of its loop groups. Our identification preserves natural module structures over the representation ring $R(G)$ and a natural duality pairing. Two earlier papers in the series covered foundations of twisted equivariant $K$-theory, introduced distinguished families of Dirac operators and discussed the special case of connected groups with free $\pi_1$. Here, we recall the earlier material as needed to make the paper self-contained. Going further, we discuss the relation to semi-infinite cohomology, the fusion product of conformal field theory, the rôle of energy and a topological Peter-Weyl theorem.
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