In this talk I will give a definition of geometric quantization of non-compact Hamiltonian torus manifolds as an equivariant index. Our construction is based on a deformation of Dirac operator along the orbits, which gives rise a localization of index to the lattice points in the moment polytope.
The localization property leads us to give natural proofs of a [Q,R]=0 and a Danilov-type formula in the non-compact setting.
This talk is based on a preprint arXiv:2001.02280.