Title: Geometric quantization of b-symplectic manifolds Show Chinese title

Title: b-对称流形的几何量子化

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. Show Chinese abstract

Abstract: 我们介绍一种针对紧致$b$-辛流形的几何量子化方法，该方法基于Atiyah-Patodi-Singer (APS) 边界值问题的指标。 我们进一步证明，b-辛流形在通常意义上具有规范的Spin-c结构，并且上述APS指标与Spin-c狄拉克算子的指标一致。 我们证明，如果流形配备了一个紧致连通李群的哈密顿作用，并且具有非零模量权，则此方法满足Guillemin-Sternberg的“量子化与约化交换”性质。 特别是我们的量子化与Guillemin、Miranda和Weitsman定义的形式量子化一致，从而对他们在论文中提出的问题给出了肯定回答。