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Article contents
Deformation of Dirac operators along orbits and quantization of noncompact Hamiltonian torus manifolds
Part of:
Partial differential equations on manifolds; differential operators
$K$-theory and operator algebras
Symplectic geometry, contact geometry
Published online by Cambridge University Press: 09 March 2021
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.
MSC classification
Primary:
19K56: Index theory
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- Copyright
- © Canadian Mathematical Society 2021
Footnotes
The author is partly supported by Grant-in-Aid for Scientific Research (C) 18K03288.
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