Loop groups and twisted 𝐾-theory II

Freed, Daniel¹ ; Hopkins, Michael² ; Teleman, Constantin³

¹ Department of Mathematics, University of Texas, 1 University Station C1200, Austin, Texas 78712-0257
² Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138
³ Department of Mathematics, University of California, 970 Evans Hall, Berkeley, California 94720-3840

Journal of the American Mathematical Society, Tome 26 (2013) no. 3, pp. 595-644

Résumé

This is the second in a series of papers investigating the relationship between the twisted equivariant $K$-theory of a compact Lie group $G$ and the “Verlinde ring” of its loop group. We introduce the Dirac family of Fredholm operators associated to a positive energy representation of a loop group. It determines a map from isomorphism classes of representations to twisted $K$-theory, which we prove is an isomorphism if $G$ is connected with a torsion-free fundamental group. We also introduce a Dirac family for finite dimensional representations of compact Lie groups; it is closely related to both the Kirillov correspondence and the equivariant Thom isomorphism. (In Part III of this series we extend the proof of our main theorem to arbitrary compact Lie groups $G$ and provide supplements in various directions. In Part I we develop twisted equivariant $K$-theory and carry out some of the computations needed here.)

Détail

DOI : 10.1090/S0894-0347-2013-00761-4

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