2020/01/07 by Hajime Fujita, Fujita, Hajime
Mathematics · #57S25 #58J22 #Differential Geometry (math.DG) #FOS: Mathematics #K-Theory and Homology (math.KT) #Primary 19K56 #Secondary 53D50 #Symplectic Geometry (math.SG) #math.DG #math.KT #math.SG #msc:19K56 #msc:53D50 #msc:57S25 #msc:58J22
paper · pdf · doi:10.48550/arxiv.2001.02280
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 Especially descriptions for non-abelian case are withdrawn.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
arxiv created 2021/03/01 · arxiv updated 2021/03/02
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.