# Spin-c Dirac operators and b-symplectic manifolds

## Meeting Details

Abstract: We introduce a method of geometric quantization for compact b-symplectic manifolds in terms of the index of an Atiyah-Patodi-Singer (APS) boundary value problem. We show furthermore that (oriented) b-symplectic manifolds have canonical Spin-c structures in the usual sense, and that the APS index above coincides with the index of the Spin-c Dirac operator. We show that if the manifold is endowed with a Hamiltonian action of a compact connected Lie group with non-zero modular weights, then this method satisfies the Guillemin-Sternberg quantization commutes with reduction'' property. In particular our quantization coincides with the formal quantization defined by Guillemin, Miranda and Weitsman, providing a positive answer to a question posed in their paper. This is joint work with M. Braverman and Y. Song.