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 $\mathrm{\Lambda }$, a 3-form $\phi$, and a closed 1-form $\theta$ such that the exterior derivative $d\phi$ of $\phi$ is the wedge product of $\theta$ and $\phi$, the anchor ${\mathrm{\Lambda }}^{\mathrm{\#}}(\theta )$ of $\theta$ is identically zero, and the Jacobiator (Jacobi operator; which is half the Schouten-Nijenhuis bracket of $\mathrm{\Lambda }$ with itself) associated to $\mathrm{\Lambda }$ is the anchor ${\mathrm{\Lambda }}^{\mathrm{\#}}(\phi )$ of $\phi$. 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.