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.

中文翻译：

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$ b $-辛流形的几何量化

我们根据Atiyah–Patodi–Singer（APS）边值问题的指数，介绍了紧凑的$ b $-辛流形的几何量化方法。我们进一步证明，在通常意义上，$ b $-辛歧形流形具有规范的$ \ operatorname {Spin} $ ‑ $ c $结构，并且上面的APS索引与$ \ operatorname {Spin} $ ‑ $的索引重合。 c $ Dirac运算符。我们表明，如果流形具有非零模块化权重的紧密连接的李群的哈密顿作用，那么该方法就满足了Guillemin-Sternberg的“量化减量化”。特别是，我们的量化与Guillemin，Miranda和Weitsman定义的形式量化相吻合，从而为他们论文中提出的问题提供了肯定的答案。