Geodesic
Deformation of Dirac operators along orbits and quantization of noncompact Hamiltonian torus manifolds
Canadian journal of mathematics, Tome 74 (2022) no. 4, pp. 1062-1092

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DOI

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.
DOI : 10.4153/S0008414X2100016X
Mots-clés : Equivariant index, geometric quantization, localization 
Fujita, Hajime. Deformation of Dirac operators along orbits and quantization of noncompact Hamiltonian torus manifolds. Canadian journal of mathematics, Tome 74 (2022) no. 4, pp. 1062-1092. doi: 10.4153/S0008414X2100016X
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     title = {Deformation of {Dirac} operators along orbits and quantization of noncompact {Hamiltonian} torus manifolds},
     journal = {Canadian journal of mathematics},
     pages = {1062--1092},
     year = {2022},
     volume = {74},
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     doi = {10.4153/S0008414X2100016X},
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