An infinite-dimensional index theorem and the
Higson–Kasparov–Trout algebra

Takata, Doman

doi:10.2140/akt.2022.7.1

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Comments and Further Discussion 61 Application To Quantizati...1

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2024

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Cited by 2 publications

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“…In [24], an LS 1 -equivariant index is constructed as an element in the fusion ring from the viewpoint of KK-theory and noncommutative geometry. He also developed an index theorem in infinitedimensional setting in [23,25]. It would be also interesting to investigate how our construction is positioned in Takata's theory.…”

Section: Comments and Further Discussion 61 Application To Quantizati...mentioning

confidence: 99%

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

Fujita

2021

Can. J. Math.-J. Can. Math.

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We give a formulation of a deformation of Dirac operator along orbits of a group action on a possibly non-compact manifold to get an equivariant index and a K-homology cycle representing the index. We apply this framework to non-compact Hamiltonian torus manifolds to define geometric quantization from the view point 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 non-compact setting, which a localization phenomenon of geometric quantization in the non-compact setting. The proofs are based on the localization of index to lattice points.

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Section: Comments and Further Discussion 61 Application To Quantizati...mentioning

confidence: 99%