发表机构
Dartmouth College（达特茅斯学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究$b^k$-辛流形的几何量子化，推广Guillemin等的工作，证明其量子化是Spin$^c$-Dirac算子的指标，有限维，且模次数为奇数时量子化与约化可交换，偶数时$[Q,R]=0$不成立。
AI 中文摘要
我们利用李代数胚的可积性研究 $b^k$-辛流形的几何量子化。借助群胚指标，我们为满足以下条件的 $b^k$-辛流形定义了一种量子化：其奇异轨迹为正规相交除子，且带有紧连通李群的哈密顿作用，这在两个方向上推广了Guillemin--Miranda--Weitsman的工作。我们证明该量子化是Spin$^c$-Dirac算子的指标，回答了他们的一个问题；特别地，对每个$k$，它都是有限维的虚表示，而当模次数为偶数时，他们的形式量子化是无限维的。最后，我们证明当模次数为奇数时，量子化与约化可交换。该假设是必要的而非技术性的：有限维指标无法与无限维形式量子化一致。我们给出了一个明确的例子，其中当模次数为偶数时，$[Q,R]=0$不成立。
英文摘要
We study the geometric quantization of $b^k$-symplectic manifolds using the integrability of Lie algebroids. Using a groupoid index, we define a quantization for $b^k$-symplectic manifolds whose singular locus is a normal crossing divisor and which carry a Hamiltonian action of a compact connected Lie group, generalizing Guillemin--Miranda--Weitsman in a few directions. Firstly, we show this quantization is the index of a $\spinc$-Dirac operator, answering a question of theirs. In particular, it is a finite-dimensional virtual representation for every $k$, whereas their formal quantization is infinite-dimensional when the modular degrees are even. Secondly, our symplectic form can have singularities along hypersurfaces which can have normal crossings. Finally, we prove that quantization commutes with reduction for the Hamiltonian action of a possibly non-abelian compact connected Lie group, when the modular degrees are odd. In the case when the modular degrees are not odd, we give an example when $[Q,R]=0$ fails.
