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arXiv 2001.022802020-01math.DGmath.KTmath.SG

Deformation of Dirac operators along orbits and quantization of non-compact Hamiltonian torus manifolds

Hajime Fujita

备注：: 27pages. Due to referee's comments several expositions are rewritten, and typos are corrected. Especially descriptions for non-abelian case are withdrawn. References uploaded. To appear in Canadian Journal of Mathematics

英文摘要：

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 shows that this geometric quantization is independent of the choice of polarization. The proofs are based on the localization of index to lattice points.

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