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arXiv · 1910.10016

Geometric quantization of b-symplectic manifolds

Maxim Braverman·Yiannis Loizides·Yanli Song
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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 further that 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.

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Maxim Braverman, Yiannis Loizides, Yanli Song. 2019-10-22. Geometric quantization of b-symplectic manifolds. https://arxiv.org/abs/1910.10016

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