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

Abstract

We introduce and investigate the concept of a $\theta$-almost twisted Poisson manifold $(M,\Lambda,\varphi,\theta)$. This structure consists of a smooth manifold $M$ equipped with a bivector field $\Lambda$, a 3-form $\varphi$, and a closed 1-form $\theta$, satisfying the following conditions: the exterior derivative $d\varphi$ of $\varphi$ equals the wedge product $\theta \wedge \varphi$; the anchor $\Lambda^\#(\theta)$ of $\theta$ vanishes identically; and one-half of the Schouten-Nijenhuis bracket $[\Lambda, \Lambda]$ equals the anchor $\Lambda^\#(\varphi)$ of $\varphi$. This structure generalizes both Poisson and twisted Poisson manifolds, permitting the 3-form $\varphi$ to be non-closed in a way controlled by the 1-form $\theta$. We construct a Lie-Rinehart algebra on the module of 1-forms $\Omega^1(M)$, giving rise to a cochain complex and an associated cohomology theory called $\theta$-almost twisted Poisson cohomology. Moreover, we develop the geometric quantization of these manifolds by defining a suitable contravariant derivative, establishing a prequantization condition in terms of the cohomology, and constructing a quantum Hilbert space via polarization. We illustrate our results with several examples, including the computation of the cohomology and quantization on $\mathbb{R}^5$.