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Freed, Daniel S.; Hopkins, Michael J.; Teleman, Constantin

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

Ann. Math. (2) 174, No. 2, 947-1007 (2011).
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.

Cited in 6 Reviews
Cited in 29 Documents

MSC:

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

Keywords:

Verlinde group; semi-infinite cohomology; conformal field theory

Citations:

Zbl 1241.19002
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\textit{D. S. Freed} et al., Ann. Math. (2) 174, No. 2, 947--1007 (2011; Zbl 1239.19002)
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References:

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