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Mathematische Annalen

, Volume 374, Issue 1–2, pp 681–722 | Cite as

Quantization of Hamiltonian loop group spaces

  • Yiannis LoizidesEmail author
  • Yanli Song
Article
  • 51 Downloads

Abstract

We prove a Fredholm property for spin-c Dirac operators \(\mathsf {D}\) on non-compact manifolds satisfying a certain condition with respect to the action of a semi-direct product group \(K\ltimes \Gamma \), with K compact and \(\Gamma \) discrete. We apply this result to an example coming from the theory of Hamiltonian loop group spaces. In this context we prove that a certain index pairing \([{\mathcal {X}}] \cap [\mathsf {D}]\) yields an element of the formal completion \(R^{-\infty }(T)\) of the representation ring of a maximal torus \(T \subset H\); the resulting element has an additional antisymmetry property under the action of the affine Weyl group, indicating \([{\mathcal {X}}] \cap [\mathsf {D}]\) corresponds to an element of the ring of projective positive energy representations of the loop group.

Notes

Acknowledgements

We especially thank Eckhard Meinrenken for many helpful discussions and encouragement, and for providing feedback on an earlier draft. We also thank Nigel Higson for helpful discussions. Y. Song is supported by NSF grant 1800667.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Pennsylvania State UniversityUniversity ParkUSA
  2. 2.Washington UniversityMissouriUSA

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