On the geometric quantization of $\theta$-almost twisted Poisson manifold

A $\theta$-almost twisted Poisson manifold is a manifold $M$ together with a bivector field $\Lambda$, a 3-form $\varphi$, and a closed 1-form $\theta$ such that the exterior derivative $d\varphi$ of $\varphi$ is the wedge product of $\theta$ and $\varphi$, the anchor $\Lambda^\#(\theta)$ of $\theta$ is identically zero, and the Jacobiator (Jacobi operator; which is half the Schouten-Nijenhuis bracket of $\Lambda$ with itself) associated to $\Lambda$ is the anchor $\Lambda^\#(\varphi)$ of $\varphi$. In this work, we define the notion of a contravariant derivative adapted to these manifolds and establish the prequantization condition in terms of the $\theta$-almost twisted Poisson cohomology. We then introduce a polarization and construct a quantum Hilbert space. These results are illustrated by examples.