Loop Groups and Twisted K-Theory II

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

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arXiv · math/0511232

Loop Groups and Twisted K-Theory II

Daniel S. Freed·Michael J. Hopkins·Constantin Teleman

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Abstract

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 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 (math.AT/0312155) we extend the proof of our main theorem to arbitrary compact Lie groups G and provide supplements in various directions. In Part I (arXiv:0711.1906) we develop twisted equivariant K-theory and carry out some of the computations needed here. We refer to the announcements math.AT/0312155 and math.AT/0206237 for further expository material and motivation.

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BibTeX RIS

Daniel S. Freed, Michael J. Hopkins, Constantin Teleman. 2012-12-07. Loop Groups and Twisted K-Theory II. https://arxiv.org/abs/math/0511232

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