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Witten Non Abelian Localization for Equivariant K-Theory, and the $[Q,R]=0$ Theorem
About this Title
Paul-Emile Paradan, Institut Montpelliérain Alexander Grothendieck, CNRS UMR 5149, Université de Montpellier, France and Michèle> Vergne, Institut de Mathématiques de Jussieu, CNRS UMR 7586, Université Paris 7, France
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 261, Number 1257
ISBNs: 978-1-4704-3522-6 (print); 978-1-4704-5397-8 (online)
DOI: https://doi.org/10.1090/memo/1257
Published electronically: November 5, 2019
Keywords: Dirac operators,
Clifford bundle,
geometric quantization,
non-abelian localization
MSC: Primary 58J20, 53D50, 53C27; Secondary 19K56, 57S15
Table of Contents
Chapters
- Introduction
- 1. Index Theory
- 2. $\mathbf {K}$-theoretic localization
- 3. “Quantization commutes with Reduction” Theorems
- 4. Branching laws
Abstract
The purpose of the present memoir is two-fold. First, we obtain a non-abelian localization theorem when M is any even dimensional compact manifold : following an idea of E. Witten, we deform an elliptic symbol associated to a Clifford bundle on M with a vector field associated to a moment map. Second, we use this general approach to reprove the [Q,R] = 0 theorem of Meinrenken-Sjamaar in the Hamiltonian case, and we obtain mild generalizations to almost complex manifolds. This non-abelian localization theorem can be used to obtain a geometric description of the multiplicities of the index of general $spin^c$ Dirac operators.- Michael Francis Atiyah, Elliptic operators and compact groups, Lecture Notes in Mathematics, Vol. 401, Springer-Verlag, Berlin-New York, 1974. MR 0482866
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