bk-Symplectic Manifolds and [Q,R]=0
Abstract
We study the geometric quantization of bk-symplectic manifolds using the integrability of Lie algebroids. Using a groupoid index, we define a quantization for bk-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.
Related
- Fedosov manifolds with parallel symplectic Weyl curvature
- Symplectic four-manifolds and hyperbolic π-orbifolds
- The ECH and alternative ECH capacities of closed symplectic 4-manifolds
- Finite Symplectic Automorphisms of Hyperkähler Manifolds of K3[n]-Type Act Trivially on Discriminant Groups
- Mazur manifolds and symplectic structures
BibTeX
@misc{sadegh2026bksymplectic,
title = {{$b^k$-Symplectic Manifolds and $[Q,R]=0$}},
author = {Ahmad Reza Haj Saeedi Sadegh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.22171},
eprint = {2608.22171},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22171},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - b^k-Symplectic Manifolds and [Q,R]=0
AU - Ahmad Reza Haj Saeedi Sadegh
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2608.22171
UR - https://arxiv.org/abs/2608.22171
AB - 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.
ER -