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

Abstract

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. H. FUJITA 7.2. [Q,R]=0 for non-compact Hamiltonian torus manifolds 28 7.3. A Danilov-type formula for non-compact toric manifolds 29 8. Comments and further discussions 30 8.1. Application to quantization of Hamiltonian loop group spaces 30 8.2. Deformation as KK-products 31 References 31