Loop groups and twisted \(K\)-theory. III. (English) Zbl 1239.19002

Summary: For part I see [the authors, J. Topol. 4, No. 4, 737–798 (2011; Zbl 1241.19002)].
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.


19L50 Twisted \(K\)-theory; differential \(K\)-theory
19L47 Equivariant \(K\)-theory
22E67 Loop groups and related constructions, group-theoretic treatment
81T45 Topological field theories in quantum mechanics


Zbl 1241.19002
Full Text: DOI


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