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Deformation of Dirac operators along orbits and quantization of noncompact Hamiltonian torus manifolds

Part of: Partial differential equations on manifolds; differential operators Symplectic geometry, contact geometry $K$-theory and operator algebras

Published online by Cambridge University Press:  09 March 2021

Hajime Fujita
Hajime Fujita*
Affiliation:
Department of Mathematical and Physical Sciences, Japan Women’s University, Tokyo, Japan
*
e-mail: fujitah@fc.jwu.ac.jp
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Abstract

We give a formulation of a deformation of Dirac operator along orbits of a group action on a possibly noncompact manifold to get an equivariant index and a K-homology cycle representing the index. We apply this framework to noncompact Hamiltonian torus manifolds to define geometric quantization from the viewpoint of index theory. We give two applications. The first one is a proof of a [Q,R]=0 type theorem, which can be regarded as a proof of the Vergne conjecture for abelian case. The other is a Danilov-type formula for toric case in the noncompact setting, which is a localization phenomenon of geometric quantization in the noncompact setting. The proofs are based on the localization of index to lattice points.

Keywords

Equivariant indexgeometric quantizationlocalization

MSC classification

Primary: 19K56: Index theory
Secondary: 53D50: Geometric quantization 57S25: Groups acting on specific manifolds 58J22: Exotic index theories
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 4 , August 2022 , pp. 1062 - 1092
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

The author is partly supported by Grant-in-Aid for Scientific Research (C) 18K03288.

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