Loop groups and twisted $K$-theory III

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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Authors

Daniel S. Freed

Department of Mathematics
The University of Texas at Austin
Austin, TX 78712

Michael J. Hopkins

Department of Mathematics
Harvard University
Cambridge, MA 02138

Constantin Teleman

Department of Mathematics
University of California, Berkeley
Berkeley, CA 94720-3840