Journal of Symplectic Geometry

Volume 19 (2021)

Number 1

Geometric quantization of $b$-symplectic manifolds

Pages: 1 – 36

DOI: https://dx.doi.org/10.4310/JSG.2021.v19.n1.a1

Authors

Maxim Braverman (Department of Mathematics, Northeastern University, Boston, Massachusetts, U.S.A.)

Yiannis Loizides (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)

Yanli Song (Department of Mathematics, Washington University, St. Louis, Missouri, U.S.A.)

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 $\operatorname{Spin}$‑$c$ structures in the usual sense, and that the APS index above coincides with the index of the $\operatorname{Spin}$‑$c$ 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.

Received 9 January 2020

Accepted 31 July 2020

Published 26 March 2021