## Loop groups and twisted $K$-theory II

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- by Daniel S. Freed, Michael J. Hopkins and Constantin Teleman
- J. Amer. Math. Soc.
**26**(2013), 595-644 - DOI: https://doi.org/10.1090/S0894-0347-2013-00761-4
- Published electronically: February 7, 2013
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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 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.)## References

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## Bibliographic Information

**Daniel S. Freed**- Affiliation: Department of Mathematics, University of Texas, 1 University Station C1200, Austin, Texas 78712-0257
- Email: dafr@math.utexas.edu
**Michael J. Hopkins**- Affiliation: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138
- Email: mjh@math.harvard.edu
**Constantin Teleman**- Affiliation: Department of Mathematics, University of California, 970 Evans Hall, Berkeley, California 94720-3840
- Email: C.Teleman@dpmms.cam.ac.uk
- Received by editor(s): November 9, 2009
- Received by editor(s) in revised form: December 7, 2012
- Published electronically: February 7, 2013
- Additional Notes: During the course of this work the first author was partially supported by NSF grants DMS-0072675 and DMS-0305505.

During the course of this work the second author was partially suppported by NSF grants DMS-9803428 and DMS-0306519

During the course of this work the third author was partially supported by NSF grant DMS-0072675

The authors also thank the KITP of Santa Barbara (NSF Grant PHY99-07949) and the Aspen Center for Physics for hosting their summer programs, where various sections of this paper were revised and completed. - © Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**26**(2013), 595-644 - MSC (2010): Primary 22E67, 57R56, 19L50
- DOI: https://doi.org/10.1090/S0894-0347-2013-00761-4
- MathSciNet review: 3037783