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

Nasser Saipele Nansidi, Bertuel Tangue Ndawa, Joseph Dongho

► To cite this version:

Nasser Saipele Nansidi, Bertuel Tangue Ndawa, Joseph Dongho. On the geometric quantization of $\theta$-almost twisted Poisson manifold. 2025. hal-05283632

HAL Id: hal-05283632

https://hal.science/hal-05283632v1

Preprint submitted on 31 Oct 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# On the geometric quantization of $\theta$-almost twisted Poisson manifold

Nasser Saipele Nansidi¹, Bertuel Tangue Ndawa²,³, and Joseph Dongho¹

¹Faculty of Science, University of Maroua, Cameroon.

²University Institute of Technology, University of Ngaoundere, Cameroon.

³Institut des Hautes études Scientifiques, Université Paris-Saclay, France.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

**Abstract** 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.

**Keywords:** $\theta$-almost twisted Poisson manifold, Geometric quantization, pre-quantization, cohomology.

## 1 Introduction

Geometric quantization, pioneered by Kostant and Souriau, offers a profound framework for constructing quantum systems from classical phase spaces modeled by symplectic or Poisson manifolds. This procedure, encompassing pre-quantization and polarization, assigns symplectic manifolds to Hilbert spaces, and classical observables to Hermitian operators. While thoroughly developed for symplectic and Poisson manifolds, recent advances in mathematical physics require generalizations to incorporate broader structures arising in string theory, noncommutative geometry, and deformation quantization. These generalizations have the potential to reveal new insights into the interplay between classical and quantum systems.
A pivotal innovation emerged with twisted Poisson manifolds [11], where a closed 3-form $\varphi$ deforms the Jacobi identity through $\frac{1}{2}[\Lambda, \Lambda] = \Lambda^{\#}(\varphi)$. These structures encapsulate topological fluxes in string theories. In [4], Pedalidou established their geometric quantization by introducing Lichnerowicz-twisted Poisson cohomology to characterize prequantization bundles, and polarization-compatible observables.

Nonetheless, physical settings in topological field theory (see[1]) motivate $\theta$-almost twisted Poisson manifolds $(M, \Lambda, \varphi, \theta)$ (see[10]), where a closed 1-form $\theta$ is linked to the 3-form $\varphi$ via $d\varphi = \theta \wedge \varphi$, and $\Lambda^{\#}(\theta) = 0$. Such manifolds generalize twisted Poisson manifolds. The non-closure of $\varphi$ in this setting disrupts conventional geometric quantization for twisted Poisson manifold. To address this challenge, we generalize Pedalidou’s framework to $\theta$-almost twisted Poisson manifolds by using adapted cohomology [10] to define $\theta$-almost twisted Poisson-Chern classes, establish prequantization condition, and construct a quantum Hilbert space.

We begin by reviewing the essential concepts in section 2 before presenting our results in sections 3,4,5.

## 2 Tools

Throughout this paper, $k \ge 1$, all objects are smooth, and $M$ denotes a manifold. The set of smooth maps on $M$ is denoted by $C^{\infty}(M)$. For $p \ge 0$, $\Omega^p(M)$ and $\mathcal{X}^p(M)$ denote the set of $p$-forms and $p$-vector fields on $M$, respectively. In particular, $\Omega^0(M) = \mathcal{X}^0(M) = C^{\infty}(M)$. We denote $\Omega(M) = \bigoplus_{p \ge 0} \Omega^p(M)$ and $\mathcal{X}(M) = \bigoplus_{p \ge 0} \mathcal{X}^p(M)$, where $\bigoplus$ denotes the direct sum.

### 2.1 Basic definitions

Let $X \in \mathcal{X}(M)$, and let $\psi \in \Omega^k(M)$. We denote by $i_X\psi$ the interior derivative of $\psi$, which is defined by

$$i_X\psi(X_1, \dots, X_{k-1}) = \psi(X, X_1, \dots, X_{k-1})$$

for every $X_1, \dots, X_{k-1} \in \Omega^1(M)$. Observe that the map $i_X : \Omega(M) \to \Omega^{k-1}(M)$ is a $C^{\infty}(M)$-linear endomorphism on $\Omega(M)$.

A bivector $\Lambda \in \mathcal{X}^2(M)$ naturally induces a morphism $\Lambda^{\#} : \Omega^k(M) \to \mathcal{X}^k(M)$ where

$$\Lambda^{\#}(\psi)(\alpha_1, \dots, \alpha_k) = (-1)^k \varphi \left( \Lambda^{\#}(\alpha_1), \dots, \Lambda^{\#}(\alpha_k) \right), \quad (2.1)$$

for every $\psi \in \Omega^k(M)$ and $\alpha_1, \dots, \alpha_k \in \Omega^1(M)$. In particular, $\Lambda^{\#}(f) = f$ for $f \in C^{\infty}(M)$, and $\Lambda^{\#}(\alpha)$ is a vector field for every $\alpha \in \Omega^1(M)$. For a differential form $\psi$ on $M$, we denote $\Lambda^{\#}(\psi)$ by $\psi^{\#}$.

### 2.2 Koszul and Schouten brackets

The Koszul bracket $[ , ]_K$ associated to a bivector field $\Lambda$ is defined by

$$[\alpha, \beta]_K = \mathcal{L}_{\Lambda^{\#}(\alpha)}(\beta) - \mathcal{L}_{\Lambda^{\#}(\beta)}(\alpha) - d\Lambda(\alpha, \beta), \quad \alpha, \beta \in \Omega^1(M).$$
The Schouten-Nijenhuis (or Schouten) bracket had been introduced by Schouten ([5]) and Nijenhuis ([2]). Its definition stems from the following result.

**Theorem 2.1.** [9, Theorem 2.8] There exists a unique $\mathbb{R}$-bilinear operation $[\cdot, \cdot] : \mathfrak{X}^{p+1}(M) \times \mathfrak{X}^{q+1}(M) \to \mathfrak{X}^{p+q+1}(M)$, $(p, q \ge -1)$ which satisfies the following properties:

(i) When $p = q = 0$, it is the usual Lie bracket of vector fields;

(ii) When $p = 0$ and $q = -1$, it is the Lie derivative $[X, f] = \mathcal{L}_X f = X(f)$;

For $P \in \mathfrak{X}^{p+1}(M)$, $Q \in \mathfrak{X}^{q+1}(M)$, and $R \in \mathfrak{X}^{r+1}(M)$,

(iii) Graded skew-symmetry: $[P, Q] = -(-1)^{pq}[Q, P]$;

(iv) Graded Leibniz identity: $[P, Q \wedge R] = [P, R] \wedge R + (-1)^{p(q+1)}Q \wedge [P, R]$;

(v) Graded Jacobi identity: the graded Jacobi operator (Jacobiator) $J_{[\cdot, \cdot]}(P, Q, R)$ of $P, Q, R$ is identically null, where

$$J_{[\cdot, \cdot]}(P, Q, R) = (-1)^{rp}[P, [Q, R]] + (-1)^{pq}[Q, [R, P]] + (-1)^{qr}[R, [P, Q]].$$

## 2.3 Lie-Rinehart algebra

Let $R$ be a commutative ring. Let $A$ be an algebra over $R$. We recall that a derivation of $A$ is a morphism $\delta: A \to A$ of $R$-modules so that $\delta(ab) = \delta(a)b + a\delta(b)$ for all $a, b \in A$. We denote by $\text{Der}(A)$ the set of derivations of $A$. It is well known that $\text{Der}(A)$ is a Lie algebra over $R$ under the commutator of derivations $[\delta_1, \delta_2] = \delta_1 \circ \delta_2 - \delta_2 \circ \delta_1$. The $R$-module $\text{Der}(A)$ is well known to be a $A$-module if the algebra $A$ is commutative.

**Definition 2.2.** [6, page 5] Let $R$ be a commutative ring, and let $A$ be a commutative $R$-algebra (not necessarily with unit). Let $(L, [-, -])$ be a Lie algebra over $R$. A Lie-Rinehart algebra structure on $L$ is a Lie algebra homomorphism $\rho: L \to \text{Der}(A)$ satisfying the following compatibility properties:

1. $\rho(a\alpha)(b) = a\rho(\alpha)(b)$,

2. $[\alpha, a\beta] = a[\alpha, \beta] + \rho(\alpha)(a)\beta$,

where $a, b \in A$ and $\alpha, \beta \in L$. A Lie-Rinehart algebra is a pair $(L, \rho)$ where $\rho$ is a Lie-Rinehart structure on $L$.

## 2.4 $\theta$-almost twisted Poisson cohomology

This part is essentially based on [10]. We first introduce $\theta$-almost twisted Poisson manifolds before presenting their associated cohomology.

### 2.4.1 $\theta$-almost twisted Poisson structures

A $\theta$-almost twisted Poisson manifold is a manifold $M$ equipped with a bivector field $\Lambda$, together with a 3-form $\varphi$ and a closed 1-form $\theta$ on $M$, such that the following hold: $d\varphi = \theta \wedge \varphi$, $\Lambda^\#(\theta) = 0$, and

$$\frac{1}{2}[\Lambda, \Lambda] = \Lambda^\#(\varphi), \qquad (2.2)$$

where $[\cdot, \cdot]$ is the Schouten-Nijenhuis bracket defined from Theorem 2.1. A $\theta$-almost twisted Poisson manifold will be denoted by $(M, \Lambda, \varphi, \theta)$.
This definition differs from that of a twisted Poisson manifold by breaking the closure of the differential 3-form $\varphi$ via the introduction of a closed differential 1-form $\theta$, while retaining specific compatibility constraints. The following examples of $\theta$-almost twisted Poisson manifolds are constructed from certain manifolds equipped with additional structures.

### Example 1.

1. Any Poisson (twisted Poisson) manifold can be endowed with a $\theta$-almost twisted Poisson structure.

2. Let $(x_1, x_2, x_3, x_4, x_5)$ be a coordinate system in $\mathbb{R}^5$. Let $\theta = dx_5$, $\Lambda = e^f \partial_{x_1} \wedge \partial_{x_2} + e^g \partial_{x_3} \wedge \partial_{x_4}$, and

$$ \varphi = \partial_{x_1} e^{-g} dx_1 \wedge dx_3 \wedge dx_4 + \partial_{x_2} e^{-g} dx_2 \wedge dx_3 \wedge dx_4 \\ + (\partial_{x_5} e^{-g} - e^{-g}) dx_3 \wedge dx_4 \wedge dx_5 + \partial_{x_3} e^{-f} dx_1 \wedge dx_2 \wedge dx_3 \\ + \partial_{x_4} e^{-f} dx_1 \wedge dx_2 \wedge dx_4 + (\partial_{x_5} e^{-f} - e^{-f}) dx_1 \wedge dx_2 \wedge dx_5, $$

where $f, g \in C^\infty(\mathbb{R}^5)$. Then the triple $(\Lambda, \varphi, \theta)$ is a $\theta$-almost twisted Poisson structure on $\mathbb{R}^5$.

### 2.4.2 $\theta$-almost twisted Poisson complex

Let $(M, \Lambda, \varphi, \theta)$ be a $\theta$-almost twisted Poisson manifold. The bracket $\{f, g\}$: $C^\infty(M) \times C^\infty(M) \to (C^\infty(M), \text{defined by } (f, g) \to \{f, g\} = \Lambda(df, dg) \text{ verifies the following equality:}}$

$$ \{f, \{g, h\}\} + \{g, \{h, f\}\} + \{h, \{f, g\}\} = \Lambda^{\#}(\varphi)(df, dg, dh). $$

As a consequence, the Chevalley-Eilenberg cohomology of what we term a $\theta$-almost $\varphi$-twisted Lie algebra $(C^\infty(M), \{, \})$ cannot be defined with respect to the representation given by Hamiltonian vector fields, that is, to the representation defined by the map $C^\infty(M) \times C^\infty(M) \to C^\infty(M)$, $(f, g) \mapsto X_f(g) = \Lambda^\#(df)(g)$.

Nevertheless, any $\theta$-almost twisted Poisson manifold $(M, \Lambda, \varphi, \theta)$ induces a Lie-Rinehart algebra (Lie algebroid) on the module of 1-forms $\Omega^1(M)$ of $M$. The associated Lie bracket on $\Omega^1(M)$ is defined by

$$ [\alpha, \beta]_{\varphi, \theta} = [\alpha, \beta]_K + i_{\Lambda^\#(\beta)} i_{\Lambda^\#(\alpha)} \varphi + \Lambda(\alpha, \beta) \cdot \theta \quad (2.3) $$

for every $\alpha, \beta \in \Omega^1(M)$.

Let $\partial_{\varphi, \theta}^n : \mathcal{X}^n(M) \to \mathcal{X}^{n+1}(M)$, $v \mapsto \partial_{\varphi, \theta}^n(v)$ be the map defined by

$$ \partial_{\varphi, \theta}^n(v)(\alpha_1, \dots, \alpha_{n+1}) = \sum_{i=1}^{n+1} (-1)^{i-1} (\alpha_i)^\# (v(\alpha_1, \dots, \hat{\alpha}_i, \dots, \alpha_{n+1})) \\ + \sum_{1 \le i < j \le n+1} (-1)^{i+j} v([\alpha_i, \alpha_j]_{\varphi, \theta}, \alpha_1, \dots, \hat{\alpha}_i, \dots, \widehat{\alpha_j}, \dots, \alpha_{n+1}) \quad (2.4) $$

for every $\alpha_1, \dots, \alpha_{n+1} \in \Omega^1(M)$. As usual, the terms with a hat symbol indicate that they are omitted. It is shown in [10] that the ordered pair $(\mathcal{X}(M), \partial_{\varphi, \theta})$ is a cochain complex.

The $\theta$-almost twisted Poisson cohomology of $(M, \Lambda, \varphi, \theta)$ is the cohomology of $(\mathcal{X}(M), \partial_{\varphi, \theta})$. It is denoted by $H_{\theta-atP}^*(M, \Lambda, \varphi, \theta)$, or simply $H_{\theta-atP}^*(M)$ if
there is no ambiguity. More precisely, for every positive integer $n$,

$$H_{\theta-atP}^{n}(M) = \frac{ker(\partial_{\varphi,\theta}^{n} : \mathcal{X}^{n}(M) \to \mathcal{X}^{n+1}(M))}{Im(\partial_{\varphi,\theta}^{n-1} : \mathcal{X}^{n-1}(M) \to \mathcal{X}^{n}(M))} \quad (2.5)$$

where $\mathcal{X}^{-1}(M) = \{0\}$. The cohomology class of any element $\mu \in ker(\partial_{\varphi,\theta}^{n} : \mathcal{X}^{n}(M) \to \mathcal{X}^{n+1}(M))$ is denoted by $[\mu]_{\varphi,\theta}$.

It is also shown in [10] that

$$\partial_{\varphi,\theta} \circ \Lambda^{\#} = -\Lambda^{\#} \circ d. \quad (2.6)$$

Thus, the following result holds.

**Proposition 2.3.** Let $(M, \Lambda, \varphi, \theta)$ be a $\theta$-almost twisted Poisson manifold. We denote by $H_{dR}^{*}(M, \mathbb{R})$ the de Rham cohomology of the $\theta$-almost twisted Poisson manifold $(M, \Lambda, \varphi, \theta)$, the cochain complex homomorphism $\Lambda^{\#} : (\Omega(M), d) \to (\mathcal{X}(M), \partial_{\varphi,\theta})$ induces a homomorphism in cohomology that we again note $\Lambda^{\#}$,

$$\Lambda^{\#} : H_{dR}^{*}(M, \mathbb{R}) \to H_{\theta-atP}^{*}(M), \quad [\mu] \mapsto [\Lambda^{\#}(\mu)]_{\varphi,\theta}. \quad (2.7)$$

Moreover, if $\Lambda$ is nondegenerate, then (2.7) is an isomorphism.

## 3 $\theta$-almost twisted Poisson-Chern class of a complex line bundle over $\theta$-almost twisted Poisson manifold

Let $(M, \Lambda, \varphi, \theta)$ be a $\theta$-almost twisted Poisson manifold, $\xi : K \to M$ a complex line bundle over $M$, $\Gamma(K)$ the space of global cross sections of $\xi : K \to M$, and $\text{End}_\mathbb{C}(\Gamma(K))$ the space of complex linear endomorphisms of $\Gamma(K)$.

**Definition 3.1.** A contravariant derivative on $\xi : K \to M$ is a $\mathbb{R}$-linear mapping $D : \Omega^1(M) \to \text{End}_\mathbb{C}(\Gamma(K))$ satisfying the following conditions:

$$D_{(f\alpha)} = fD_{\alpha}, \quad \text{and} \quad D_{\alpha}(fs) = fD_{\alpha}(s) + \Lambda^{\#}(\alpha)(f)s \quad (3.1)$$

for every $\alpha \in \Omega^1(M)$, $f \in C^\infty(M)$, and $s \in \Gamma(K)$.

Let $(M, \Lambda, \varphi, \theta)$ be a $\theta$-almost twisted Poisson manifold, and $h$ a Hermitian metric on a complex line bundle $\xi : K \to M$. We say that a contravariant derivative $D$ on $\xi : K \to M$ is compatible with $h$ if

$$\Lambda^{\#}(\alpha)(h(s_1, s_2)) = h(D_{\alpha}s_1, s_2) + h(s_1, D_{\alpha}s_2) \quad (3.2)$$

for every $\alpha \in \Omega^1(M)$, and $s_1, s_2 \in \Gamma(K)$.

**Definition 3.2.** Let $\xi : K \to M$ be a complex line bundle over a $\theta$-almost twisted Poisson manifold $(M, \Lambda, \varphi, \theta)$, and $D$ a contravariant derivative on $\xi : K \to M$. The curvature of $D$ is the mapping $C_D : \Omega^1(M) \times \Omega^1(M) \to \text{End}_\mathbb{C}(\Gamma(K))$, $(\alpha, \beta) \mapsto C_D(\alpha, \beta)$ where

$$C_D(\alpha, \beta) = D_{\alpha} \circ D_{\beta} - D_{\beta} \circ D_{\alpha} - D_{[\alpha, \beta]_{\varphi, \theta}}. \quad (3.3)$$
From definition 3.1, and relation (3.3), we deduce that the curvature $C_D$ of $D$ is bilinear over $C^\infty(M)$, and skew-symmetric. Thus, from the fact that $\xi: K \to M$ is a complex line bundle over $M$, there exists a globally defined complex bivector field $P_{C_D} = P_{C_{D_1}} + iP_{C_{D_2}}$ on $M$, with $P_{C_{D_1}}, P_{C_{D_2}} \in \mathcal{X}^2(M)$ such that

$$C_D(\alpha, \beta)(s) = P_{C_D}(\alpha, \beta)s \qquad (3.4)$$

for every $\alpha, \beta \in \Omega^1(M)$, and $s \in \Gamma(K)$.

The cohomology operator $\partial_{\varphi,\theta}$ defined in (2.4) can be extended by linearity to the module $\mathcal{X}_C(M)$ of complex multivectors on $M$. This extension is explicitly defined by setting for any $P = P_1 + iP_2 \in \mathcal{X}_C(M)$

$$\partial_{\varphi,\theta}P = \partial_{\varphi,\theta}P_1 + i\partial_{\varphi,\theta}P_2$$

where $P_1, P_2 \in \mathcal{X}(M)$. We show that $\partial_{\varphi,\theta}^2 = 0$. Therefore, we obtain the corresponding cohomology which will be called the *complex $\theta$-almost twisted Poisson cohomology*, and will be denoted by $H_{\mathcal{C}^\theta-\text{at}P}^*(M)$. Moreover, we have the following.

**Theorem 3.3.** Let $\xi: K \to M$ be a complex line bundle over a $\theta$-almost twisted Poisson manifold $M$, $D$ a contravariant derivative on $\xi: K \to M$, $C_D$ the curvature of $D$, and $P_{C_D}$ the complex bivector field associated to $C_D$ (3.4). Then:

(i) $P_{C_D}$ define a cohomology class $[P_{C_D}]_{\varphi,\theta}$ in $H_{\mathcal{C}^\theta-\text{at}P}^2(M)$;

(ii) $[P_{C_D}]_{\varphi,\theta}$ does not depend of the contravariant derivative $D$;

(ii) If $D$ is compatible with a Hermitian metric $h$ on $\xi: K \to M$, then $P_{C_D}$ is purely imaginary.

*Proof.* (i) Let $s$ be a local nowhere vanishing section. The bundle $\xi: K \to M$ being complex 1-dimensional, for every 1-form $\alpha$ on $M$, $\frac{D_\alpha s}{s}$ is a complex number. From (3.1), the function $\alpha \mapsto \frac{D_\alpha s}{s}$ is $\mathbb{R}$-linear, and $C^\infty(M)$-linear. Then, there exists a unique complex local vector field $X = X_1 + iX_2$ with $X_1, X_2 \in \mathcal{X}^1(M)$ such that

$$D_\alpha s = X(\alpha)s, \quad \forall \alpha \in \Omega^1(M). \qquad (3.5)$$

By using (3.3), (3.5), and (2.4), we get:

$$
\begin{align*}
C_D(\alpha, \beta)(s) &= (D_\alpha D_\beta - D_\beta D_\alpha - D_{[\alpha, \beta]_{\varphi, \theta}})(s) \\
&= D_\alpha(X(\beta)s) - D_\beta(X(\alpha)s) - X(\alpha, \beta)_{\varphi, \theta}s \\
&= X(\beta)D_\alpha s + \Lambda^\#(\alpha)(X(\beta))s - X(\alpha)D_\beta s \\
&\quad - \Lambda^\#(\beta)(X(\alpha))s - X(\alpha, \beta)_{\varphi, \theta}s \\
&= [\Lambda^\#(\alpha)(X(\beta)) - \Lambda^\#(\beta)(X(\alpha)) - X(\alpha, \beta)_{\varphi, \theta}] s \\
&= \partial_{\varphi, \theta}X(\alpha, \beta)s.
\end{align*}
$$

From (3.4), we have

$$P_{C_D} = \partial_{\varphi,\theta}X. \qquad (3.6)$$
Therefore, $\partial_{\varphi,\theta} P_{C_D} = \partial_{\varphi,\theta}^2(X) = 0$. Hence, $P_{C_D}$ defines a cohomology class $[P_{C_D}]_{\varphi,\theta} \in H_{C\theta-atP}^*(M)$.

(ii) Let $\tilde{D}$ be another contravariant derivative on $\xi: K \to M$ with associated vector field $\tilde{X}$. That is, $P_{C\tilde{D}} = \partial_{\varphi,\theta}\tilde{X}$. We Define $\tilde{X} = \tilde{X} - X$. Then, $P_{C\tilde{D}} = \partial_{\varphi,\theta}\tilde{X} = \partial_{\varphi,\theta}(X + \tilde{X}) = P_{C_D} + \partial_{\varphi,\theta}\tilde{X}$. Thus, $P_{C\tilde{D}}$ and $P_{C_D}$ differ by a coboundary, so $[P_{C\tilde{D}}]_{\varphi,\theta} = [P_{C_D}]_{\varphi,\theta}$.

(iii) Let $e$ be a local orthonormal basis of $\Gamma(K)$. The Compatibility of $D$ with a Hermitian metric $h$ implies that

$$ 
\begin{aligned}
0 &= \Lambda^\#(\alpha)(h(e, e)) \\
&= h(D_\alpha e, e) + h(e, D_\alpha e) \\
&= h(X(\alpha)e, e) + h(e, X(\alpha)e) \\
&= X(\alpha) + \bar{X}(\alpha).
\end{aligned}
 $$

Thus, $X$ is purely imaginary. Therefore, $P_{C_D} = \partial_{\varphi,\theta}X$ is purely imaginary. $\square$

We get the following definition.

**Definition 3.4.** Let $\xi: K \to M$ be a complex line bundle over a $\theta$-almost twisted Poisson manifold $(M, \Lambda, \varphi, \theta)$. Suppose that $D$ is a contravariant derivative with curvature $C_D$ such that $P_{C_D}$ is purely imaginary. Then the cohomology class $[\frac{i}{2\pi}P_{C_D}]_{\varphi,\theta}$ in $H_{\theta-atP}^2(M)$ will be called the first real $\theta$-almost twisted Poisson-Chern class of $\xi: K \to M$.

We recall that for a complex line bundle $\xi: K \to M$ over a differential manifold $M$, and $\nabla$ a Hermitian connection on $\xi: K \to M$, there exists a purely imaginary closed 2-form $\Omega_\nabla$ such that

$$ C_{\nabla}(X, Y)(s) = \Omega_{\nabla}(X, Y).s \quad (3.7) $$

for all $X, Y \in \mathcal{X}^1(M)$, and $s \in \Gamma(K)$ where $C_\nabla$ is the curvature of the connection $\nabla$.

The real Chern class $c_1(K, \mathbb{R}) \in H_{dR}^2(M, \mathbb{R})$ is the integral cohomology class $[\frac{i}{2\pi}\Omega_\nabla]$, see [3]. Note that the canonical injection $i: \mathbb{Z} \to \mathbb{R}$ induces a homomorphism $i: H_{dR}^2(M, \mathbb{Z}) \to H_{dR}^2(M, \mathbb{R})$. A class $[\alpha] \in H_{dR}^2(M, \mathbb{R})$ is called integral if it lies in the image of $i$, see [3].

The following result shows the relation between the first real Chern class and the first real $\theta$-almost twisted Poisson-Chern class of a complex line bundle over a $\theta$-almost twisted Poisson manifold.

**Theorem 3.5.** Let $\xi: K \to M$ be a complex line bundle over a $\theta$-almost twisted Poisson manifold $(M, \Lambda, \varphi, \theta)$, $\nabla$ a Hermitian connection on $\xi: K \to M$, and $D$ the associated Hermitian contravariant derivative to $\nabla$ on $\xi: K \to M$, that is for any $\alpha \in \Omega^1(M)$, $D_\alpha = \nabla_{\Lambda^\#(\alpha)}$. If $c_1(K, \mathbb{R})$, and $[\frac{i}{2\pi}P_{C_D}]_{\varphi,\theta}$ are respectively the first real Chern class and the first real $\theta$-almost twisted Poisson-Chern class of $\xi: K \to M$, then

$$ \Lambda^\#(c_1(K, \mathbb{R})) = \left[ \frac{i}{2\pi} P_{C_D} \right]_{\varphi,\theta} \quad (3.8) $$

where $\Lambda^\# : H_{dR}^*(M, \mathbb{R}) \to H_{\theta-atP}^*(M)$ is the homomorphism defined in (2.7).
Proof. Let $\Omega_{\nabla}$ be a purely imaginary, and closed 2-form associated to the connection $\nabla$. That is,

$$C_{\nabla}(X, Y)(s) = \Omega_{\nabla}(X, Y)s, \quad \forall X, Y \in \mathfrak{X}^1(M), s \in \Gamma(K).$$

So $c_1(K, \mathbb{R}) = [\frac{i}{2\pi}\Omega_{\nabla}] \in H_{dR}^2(M, \mathbb{R})$. For a local section $s$ of $\xi: K \to M$, we have $\nabla_Y s = \omega(Y)s$ where $\omega$ is the 1-form such that $\Omega_{\nabla} = d\omega$, see [13]. Then,

$$D_{\alpha}s = \nabla_{\Lambda^{\#}(\alpha)}s = \omega(\Lambda^{\#}(\alpha))s = -\Lambda^{\#}(\omega)(\alpha)s.$$

Thus, the associated vector field for $D$ is $X = -\Lambda^{\#}(\omega)$.

From (3.6), $P_{C_D} = \partial_{\varphi,\theta}X = -\partial_{\varphi,\theta}(\Lambda^{\#}(\omega))$. And from (2.6), $P_{C_D} = -(-\Lambda^{\#}(d\omega)) = \Lambda^{\#}(\Omega_{\nabla})$. Therefore:

$$\left[ \frac{i}{2\pi} P_{C_D} \right]_{\varphi,\theta} = \left[ \frac{i}{2\pi} \Lambda^{\#}(\Omega_{\nabla}) \right]_{\varphi,\theta} = \Lambda^{\#} \left( \left[ \frac{i}{2\pi} \Omega_{\nabla} \right] \right) = \Lambda^{\#}(c_1(K, \mathbb{R})).$$

□

# 4 Prequantization

The prequantization step in geometric quantization aims to construct a faithful representation of the classical algebra of observables as operators acting on sections of a complex line bundle. Specifically, for a $\theta$-almost twisted Poisson manifold $(M, \Lambda, \varphi, \theta)$, we seek to associate to each smooth function $f \in C^{\infty}(M)$ a Hermitian operator $\hat{f}$ on the space of cross-sections $\Gamma(K)$ of a Hermitian line bundle $\xi: K \to M$ such that the map $f \mapsto \hat{f}$ satisfies $\{f, g\} = \{\hat{f}, \hat{g}\}_{\varphi,\theta}$, where $\{\cdot, \cdot\}_{\varphi,\theta}$ is a convenient $\theta$-almost $\varphi$-twisted Lie bracket defined on $\Gamma(K)$. This requires the construction of the line bundle $\xi: K \to M$ having a specific contravariant derivative $D$.

Let $(M, \Lambda, \varphi, \theta)$ be a $\theta$-almost twisted Poisson manifold, and $D$ a contravariant derivative on a complex line bundle $\xi: K \to M$. Let $\hat{f}$ be the representation of $f \in C^{\infty}(M)$ in $\text{End}_{\mathbb{C}}(\Gamma(K))$ defined by

$$\hat{f}(s) = D_{df}s + 2\pi i f s, \quad s \in \Gamma(K). \qquad (4.1)$$

Let $A$ be the subset of $\text{End}_{\mathbb{C}}(\Gamma(K))$ defined by $A = \{ \hat{f} \in \text{End}_{\mathbb{C}}(\Gamma(K)) / f \in C^{\infty}(M) \}$. We define on $A$ the following bracket

$$\{\hat{f}, \hat{g}\}_{\varphi,\theta} = [\hat{f}, \hat{g}] - D_{i_{\Lambda^{\#}(\beta)}i_{\Lambda^{\#}(\alpha)}}\varphi - \Lambda(df, dg)D_{\theta} \qquad (4.2)$$

for every $f, g \in C^{\infty}(M)$ where $[\hat{f}, \hat{g}] = \hat{f} \circ \hat{g} - \hat{g} \circ \hat{f}$. We have the following Proposition.

**Proposition 4.1.** The hat representation $\hat{\omega}: (C^{\infty}(M), \{\cdot, \cdot\}) \to \text{End}_{\mathbb{C}}(\Gamma(K))$, $f \mapsto \hat{f}$ is a homomorphism, that is

$$\{\hat{f}, \hat{g}\} = \{\hat{f}, \hat{g}\}_{\varphi, \theta}, \quad f, g \in C^{\infty}(M). \qquad (4.3)$$

This means

$$C_D(df, dg) = -2\pi i \{f, g\}. \qquad (4.4)$$
*Proof.* From equality (4.1), and Definition 3.1, we get

$$[\hat{f}, \hat{g}] = \hat{f} \circ \hat{g} - \hat{g} \circ \hat{f} \\ = D_{df} \circ D_{dg} - D_{dg} \circ D_{df} + 4\pi i \{f, g\}. \qquad (4.5)$$

By using, (4.1), (2.3), (3.3), and (4.5), we have:

$$\begin{aligned} \{f,g\} s &= D_{d\{f,g\}} s + 2\pi i \{f,g\} s \\ &= D_{[df,dg]_{\varphi,\theta}} s - D_{i_{\Lambda^{\#(dg)}} i_{\Lambda^{\#(df)}}} \varphi s - \Lambda(df,dg) D_{\theta} s + 2\pi i \{f,g\} s \\ &= D_{df} \circ D_{dg}s - D_{dg} \circ D_{df}s - C_D(df,dg)s - D_{i_{\Lambda^{\#(dg)}} i_{\Lambda^{\#(df)}}} \varphi s \\ &\quad - \Lambda(df,dg) D_{\theta}s + 2\pi i \{f,g\} s \\ &= -C_D(df,dg)s + [\hat{f},\hat{g}] - 4\pi i \{f,g\} s - D_{i_{\Lambda^{\#(dg)}} i_{\Lambda^{\#(df)}}} \varphi s \\ &\quad - \Lambda(df,dg) D_{\theta}s + 2\pi i \{f,g\} s \\ &= \{\hat{f},\hat{g}\}_{\varphi,\theta} s - C_D(df,dg)s - 2\pi i \{f,g\} s. \end{aligned}$$

And the proposition follows.

<figure><img src="image_2.png" /></figure>

**Definition 4.2.** We say that a $\theta$-almost twisted Poisson manifold $(M, \Lambda, \varphi, \theta)$ is prequantizable if there exists a Hermitian complex line bundle $\xi: K \to M$, the prequantization bundle, such that the representation operator defined in (4.1) holds on $\Gamma(K)$ and satisfy (4.3).

According to Definition 4.2, the prequantization problem of a $\theta$-almost twisted Poisson manifold $(M, \Lambda, \varphi, \theta)$ has a solution if and only if, there exists a Hermitian complex line bundle $\xi: K \to M$ endowed with a contravariant derivative $D$ whose curvature $C_D$ satisfies

$$C_D = -2\pi i \Lambda. \qquad (4.6)$$

Therefore, the curvature $C_D$ is purely imaginary. This implies that $D$ is compatible with the Hermitian complex line bundle $\xi: K \to M$.

**Theorem 4.3.** A $\theta$-almost twisted Poisson manifold $(M, \Lambda, \varphi, \theta)$ is prequantizable if and only if there exists a vector field $Z$ on $M$, and a closed 2-form $\eta$ on $M$ representing an integral cohomology class of $M$ satisfying

$$\Lambda + \partial_{\varphi,\theta}(Z) = \Lambda^{\#}(\eta). \qquad (4.7)$$

*Proof.* Suppose that the $\theta$-almost twisted Poisson manifold $(M, \Lambda, \varphi, \theta)$ is prequantizable. Then, there exists a Hermitian complex line bundle $\xi: K \to M$ endowed with a contravariant derivative $D$ whose curvature $C_D$ satisfies $C_D = -2\pi i \Lambda$. Thus, $P_{C_D} = -2\pi i \Lambda$.

Let $\nabla$ be a Hermitian connection on $\xi: K \to M$ with curvature $\Omega_{\nabla}$, and $\tilde{D}_{\alpha} = \nabla_{\Lambda^{\#(\alpha)}}$. By Theorems 3.3, we have $[P_{C_{\tilde{D}}}]_{\varphi,\theta} = [P_{C_D}]_{\varphi,\theta}$. So, there exists a vector field $W$ such that $P_{C_{\tilde{D}}} = P_{C_D} + \partial_{\varphi,\theta} W$.

By Theorem 3.5 and Proposition 2.7, we get $\Lambda^{\#}(\Omega_{\nabla}) = P_{C_{\tilde{D}}}$. Therefore,

$$\Lambda^{\#}(\Omega_{\nabla}) = P_{C_D} + \partial_{\varphi,\theta} W = -2\pi i \Lambda + \partial_{\varphi,\theta} W.$$
By setting $Z = \frac{i}{2\pi}W$ and $\eta = \frac{i}{2\pi}\Omega_\nabla$ which is real, closed, and integral, we obtain

$$ \Lambda + \partial_{\varphi, \theta}(Z) = \Lambda^{\#}(\eta). $$

Conversely, let $Z$ be a vector field and $\eta$ be a closed 2-form on $(M, \Lambda, \varphi, \theta)$ such that (4.7). There exists Hermitian line bundle $K$ with connection $\nabla$ such that $\Omega_\nabla = -2\pi i\eta$. We define a contravariant derivative for all $\alpha \in \Omega^1(M), s \in \Gamma(K)$ as follows:

$$ D_{\alpha}s = \nabla_{\Lambda^{\#}(\alpha)}s + 2\pi i \alpha(Z)s. $$

Then,

$$
\begin{align*}
C_D(\alpha, \beta)(s) &= D_\alpha \circ D_\beta s - D_\beta \circ D_\alpha s - D_{[\alpha, \beta]_{\varphi, \theta}} s \\
&= D_\alpha \left( \nabla_{\Lambda^\#(\beta)} s + 2\pi i \beta(Z)s \right) - D_\beta \left( \nabla_{\Lambda^\#(\alpha)} s + 2\pi i \alpha(Z)s \right) \\
&\quad - \nabla_{\Lambda^\#([\alpha, \beta]_{\varphi, \beta})} s - 2\pi i [\alpha, \beta]_{\varphi, \beta}(Z)s \\
&= \nabla_{\Lambda^\#(\alpha)} \left( \nabla_{\Lambda^\#(\beta)} s + 2\pi i \beta(Z)s \right) + 2\pi i \alpha(Z) \left( \nabla_{\Lambda^\#(\alpha)} s + 2\pi i \beta(Z)s \right) \\
&\quad - \nabla_{\Lambda^\#(\beta)} \left( \nabla_{\Lambda^\#(\alpha)} s + 2\pi i \alpha(Z)s \right) - 2\pi i \beta(Z) \left( \nabla_{\Lambda^\#(\alpha)} s + 2\pi i \alpha(Z)s \right) \\
&\quad - \nabla_{[\Lambda^\#(\alpha), \Lambda^\#(\beta)]} s - 2\pi i [\alpha, \beta]_{\varphi, \beta}(Z)s \\
&= \Omega_\nabla (\Lambda^\#(\alpha), \Lambda^\#(\beta))s + 2\pi i \Lambda^\#(\alpha)(\beta(Z))s \\
&\quad - 2\pi i \left[ \Lambda^\#(\beta)(\alpha(Z)) + [\alpha, \beta]_{\varphi, \beta}(Z) \right]s \\
&= -2\pi i \eta (\Lambda^\#(\alpha), \Lambda^\#(\beta))s + 2\pi i \partial_{\varphi, \theta} Z(\alpha, \beta)s \\
&= -2\pi i \left[ \Lambda^\#(\eta)(\alpha, \beta) - \partial_{\varphi, \theta} Z(\alpha, \beta) \right]s \\
&= -2\pi i \Lambda(\alpha, \beta)s.
\end{align*}
$$

Given that the vector field $Z$ is real, the contravariant derivative $D$ is Hermitian. We conclude that it satisfies the prequantization condition. The following remark gives rise to a large variety of classes of prequantizable $\theta$-almost twisted Poisson manifold. $\square$

**Remark 4.4.**

1. Let $(M, \Lambda, \varphi)$ be a twisted Poisson manifold. From Theorem 4.3, it follows that $M$ is prequantizable as a 0-almost twisted Poisson manifold if and only if there exists a vector field $Z$ and a closed 2-form $\eta$ representing an integral cohomology class of $M$ satisfying (4.7). Conversely, a 0-almost twisted Poisson manifold $(M, \Lambda, \varphi, O)$ is prequantizable if and only if the triplet $(M, \Lambda, \varphi)$ is prequantizable as a twisted Poisson manifold. Indeed, $\partial_{\varphi,0} = \partial_{\varphi}$.

2. Let $(M_0, \Lambda_0, \varphi_0)$ be a twisted Poisson manifold. Consider the $\theta$-almost twisted Poisson structure on $M = M_0 \times \mathbb{R}$ defined by $\Lambda = e^t \Lambda_0, \varphi = e^{-t} \varphi_0$, and $\theta = -dt$ where $t$ is the coordinate

(a) Assume that $\Lambda_0$ is nondegenerate. Then the induced $\theta$-almost twisted Poisson manifold $(M, \Lambda, \varphi, \theta)$ is not prequantizable. Let's prove that the prequantization equation (4.7) has no solutions on $M$. Note that for any vector field $Z$ on $M_0$, there exists a differential 1-form $\alpha$
such that $Z = \Lambda_0^\#(\alpha)$. Let $\omega$ be a differential 2-form such that $\Lambda_0 = \Lambda_0^\#(\omega)$. Therefore,

$$ \Lambda + \partial_{\varphi,\theta}(Z) = \Lambda^\#(\eta) \quad (4.8) $$

where $\eta = e^{-t}\omega - d(e^{-t}\alpha)$. The differential 2-form $\eta$ cannot be identically null. Indeed, suppose it identically null. One deduces that $\omega = -dt \wedge \alpha + d\alpha$. This contradicts the fact that $\omega$ belongs to $\Omega^2(M_0)$. Observe that $\eta$ is not closed. It follows that $(M, \Lambda, \varphi, \theta)$ is not prequantizable.

(b) Assume that the twisted Poisson manifold $(M_0, \Lambda_0, \varphi_0)$ is exact in the sense of Petalidou. Then the induced $\theta$-almost twisted Poisson manifold $(M, \Lambda, \varphi, \theta)$ is prequantizable. In fact, if the twisted Poisson manifold $(M_0, \Lambda_0, \varphi_0)$ is exact, then there exists a vector field $Z_0$ on $M_0$ such that $\Lambda_0 = \partial_{\varphi_0}(Z_0)$. By identifying $M_0$ with the submanifold $\{0\} \times M_0$ of $M$, we can identify $Z_0$ as a vector field on $M$. Thus, we obtain $\partial_{\varphi,\theta}Z_0 = e^t\partial_\varphi Z_0$. Hence, by setting $Z = -Z_0$, and $\eta = 0$ which represents the integral cohomology class $[0] \in H_{dR}^2(M, \mathbb{R})$, we have the following equality:

$$ \Lambda + \partial_{\varphi,\theta}(Z) = e^t(\Lambda_0 + \partial_{\varphi_0}(Z)) = 0 = \Lambda^\#(\eta). $$

This justifies that $(M, \Lambda, \varphi, \theta)$ is prequantizable.

3. An exact $\theta$-almost twisted Poisson manifold is a $\theta$-almost twisted Poisson manifold $(M, \Lambda, \varphi, \theta)$ such that there exists a vector field $X$ on $M$ satisfying $\Lambda = \partial_{\varphi,\theta}(X)$. Thus, any exact $\theta$-almost twisted Poisson manifold is prequantizable by setting $Z = -X$, and $\eta = 0$.

4. Consider the $\theta$-almost twisted Poisson structure $(\Lambda, \varphi, \theta)$ on $\mathbb{R}^5$ defined by $\Lambda = e^f \partial_{x_1} \wedge \partial_{x_2} + e^g \partial_{x_3} \wedge \partial_{x_4}$, $\theta = dx_5$, and $\varphi = -\partial_{x_1}(g)e^{-g}dx_1 \wedge dx_3 \wedge dx_4 - \partial_{x_2}(g)e^{-g}dx_2 \wedge dx_3 \wedge dx_4 - e^{-g}(\partial_{x_5}g - 1)dx_3 \wedge dx_4 \wedge dx_5 - \partial_{x_3}(f)e^{-f}dx_1 \wedge dx_2 \wedge dx_3 - \partial_{x_4}(f)e^{-f}dx_1 \wedge dx_2 \wedge dx_4 - e^{-f}(\partial_{x_5}f - 1)dx_1 \wedge dx_2 \wedge dx_5$ where $f, g \in C^\infty(\mathbb{R}^5)$, and $(x_1, x_2, x_3, x_4, x_5)$ is a coordinate system in $\mathbb{R}^5$. Assume that $f(x_1, x_2, x_3, x_4, x_5) = f(x_1, x_2)$ and $g(x_1, x_2, x_3, x_4, x_5) = g(x_3, x_4)$. Then, $(\mathbb{R}^5, \Lambda, \varphi, \theta)$ is prequantizable. Indeed, if we take $Z = \partial_{x_5}$, and $\eta = -e^{-f}\partial_{x_1} \wedge \partial_{x_2} - e^{-g}\partial_{x_3} \wedge \partial_{x_4}$, the differential 2-form $\eta$ is closed. Since the manifold $\mathbb{R}^5$ is contractible, then, $H_{dR}^2(\mathbb{R}^5) = 0$. Therefore, the closed 2-form $\eta$ is exact and represents the trivial cohomology class, which is integral. A calculation shows that a prequantization equation (4.7) holds.

# 5 Construction of Hilbert space

In this section, we address the second step of the geometric quantization process for $\theta$-almost twisted Poisson manifolds: The construction of a Hilbert space from the prequantization space $\Gamma(K)$, on which a suitable $\theta$-almost $\varphi$-twisted Lie subalgebra of observables will be represented. We introduce the notion of polarization adapted to this setting and construct the quantum Hilbert space by tensoring the prequantization line bundle with the bundle of complex half-densities.
Let $(M, \Lambda, \varphi, \theta)$ be a $\theta$-almost twisted Poisson manifold. We consider the complexification $\Omega_{\mathbb{C}}^1(M) = \Omega^1(M) \otimes \mathbb{C}$ of $\Omega^1(M)$, and extend both the bracket $[\cdot, \cdot]_{\varphi, \theta}$, and anchor map $\Lambda^\#$ to complex-valued forms. Then, the pair $(\Omega_{\mathbb{C}}^1(M), [\cdot, \cdot]_{\varphi, \theta})$ is a complex Lie algebra. We will define a *polarization* to be a complex Lie subalgebra $\mathcal{P}$ of $(\Omega_{\mathbb{C}}^1(M), [\cdot, \cdot]_{\varphi, \theta})$ such that

$$ \Lambda(\alpha, \beta) = 0 \text{ for every } \alpha, \beta \in \Omega_{\mathbb{C}}^1(M). $$

Given such a polarization $\mathcal{P}$, we set

$$ P(\mathcal{P}) = \{f \in C^\infty(M) \mid [df, \alpha]_{\varphi, \theta} \in \mathcal{P}, \forall \alpha \in \mathcal{P}\} \text{ and } (P(\mathcal{P}))^2 = P(\mathcal{P}) \times P(\mathcal{P}). $$

Let $\Delta = \Delta(P(\mathcal{P}) \times P(\mathcal{P})) = \{(f, f), f \in P(\mathcal{P})\}$ and

$$ \widetilde{P(\mathcal{P})} = \{(f,g) \in (P(\mathcal{P}))^2 \setminus \Delta, [i_{(dg)}\# i_{(df)}\#\varphi + \Lambda(df, dg)\theta, \alpha]_{\varphi, \theta} \in \mathcal{P}, \forall \alpha \in \mathcal{P}\}. $$

Note that $\widetilde{P(\mathcal{P})}$ is symmetric with respect to $\Delta(P(\mathcal{P}) \times P(\mathcal{P}))$. We denote by $\mathcal{Q}(\mathcal{P})$, the projection of $\widetilde{P(\mathcal{P})}$ onto $P(\mathcal{P})$.

**Proposition 5.1.** The pair $(\mathcal{Q}(\mathcal{P}), \{\cdot, \cdot\})$ is a $\theta$-almost $\varphi$-twisted Lie subalgebra of $(C^\infty(M), \{\cdot, \cdot\})$.

*Proof.* Let $f, g \in \mathcal{Q}(\mathcal{P})$. It suffices to show that $\{f, g\} \in \mathcal{Q}(\mathcal{P})$, meaning $\{f, g\} \in P(\mathcal{P})$, and there exist $h \in \mathcal{P}$ with $h \neq \{f, g\}$ such that $(\{f, g\}, h) \in \widetilde{P(\mathcal{P})}$.

Let us first prove that $\{f, g\} \in P(\mathcal{P})$. By the definition of $[\cdot, \cdot]_{\varphi, \theta}$, we have

$$ d\{f, g\} = [df, dg]_{\varphi, \theta} - (i_{\Lambda^\#(dg)} i_{\Lambda^\#(df)}\varphi + \Lambda(df, dg)\theta). \quad (5.1) $$

Then by linearity, we obtain

$$ [d\{f, g\}, \alpha]_{\varphi, \theta} = [[df, dg]_{\varphi, \theta}, \alpha]_{\varphi, \theta} - [i_{\Lambda^\#(dg)} i_{\Lambda^\#(df)}\varphi + \Lambda(df, dg)\theta, \alpha]_{\varphi, \theta} $$

for every $\alpha \in \mathcal{P}$. If $f = g$, then $\{f, g\} = 0 \in P(\mathcal{P})$. Assume that $f \neq g$. Then by construction, $(f, g) \in \widetilde{P(\mathcal{P})}$. Therefore, $[i_{\Lambda^\#(dg)} i_{\Lambda^\#(df)}\varphi + \Lambda(df, dg)\theta, \alpha]_{\varphi, \theta} \in \mathcal{P}$. By using the Jacobi identity of $[\cdot, \cdot]_{\varphi, \theta}$, we get

$$ [[df, dg]_{\varphi, \theta}, \alpha]_{\varphi, \theta} = [df, [dg, \alpha]_{\varphi, \theta}]_{\varphi, \theta} + [dg, [\alpha, df]_{\varphi, \theta}]_{\varphi, \theta}. $$

Since $f, g \in P(\mathcal{P})$, and $\mathcal{P}$ is a Lie subalgebra, $[[df, dg]_{\varphi, \theta}, \alpha]_{\varphi, \theta} \in \mathcal{P}$. Therefore, $\{f, g\} \in P(\mathcal{P})$.

Note that for every constant function $h \in C^\infty(M)$, we have $[i_{\Lambda^\#(dh)} i_{\Lambda^\#(d\{f,g\})}\varphi + \Lambda(d\{f,g\}, dh)\theta, \alpha]_{\varphi, \theta} = [0, \alpha]_{\varphi, \theta} \in \mathcal{P}$. Thus, $(\{f,g\}, h) \in \widetilde{P(\mathcal{P})}$. Therefore, $\{f,g\} \in \mathcal{Q}(\mathcal{P})$. $\square$

The elements of $\mathcal{Q}(\mathcal{P})$ will be called the *straightforwardly quantizable observables* of $(M, \Lambda, \varphi, \theta)$. This extends that given in [4].

Assume $(M, \Lambda, \varphi, \theta)$ is a prequantizable $\theta$-almost twisted Poisson manifold with prequantization bundle $\xi: K \to M$, Hermitian metric $h$, and compatible contravariant derivative $D$ satisfying $C_D = -2\pi i\Lambda$. Let $\mathcal{D}$ be the half-density bundle associated to a tangent bundle $TM$. Its cross sections $\varrho$ are called *half-density* of $M$, and are complex valued map on bases of $\Gamma(TM)$ satisfying:

$$ \varrho_x(e_x A_x) = \varrho_x(e_x) | \det A_x|^{1/2} $$
for any $x \in M$, basis $e_x$ of $T_x M$, and $A_x \in \text{GL}(T_x M)$. Due to the fact that $\text{GL}(T_x M)$ acts transitively on the set of basis of $T_x M$, The bundle $\mathcal{D}$ is a complex line bundle. The Lie derivative $\mathcal{L}$ of $\varrho$ is defined as for tensor field on $M$. See [12, 8] for more details.

By using (3.1), and the properties of Lie derivative $\mathcal{L}$, we can extend the contravariant derivative $D$ to $\text{End}_\mathbb{C}(\Gamma(K \otimes \mathcal{D}))$ as follow

$$ D_{\alpha}(s \otimes \varrho) = (D_{\alpha}s) \otimes \varrho + s \otimes \mathcal{L}_{\Lambda^{\#}(\alpha)}\varrho, \quad \alpha \in \Omega_{\mathbb{C}}^{1}(M) \quad s \otimes \varrho \in \Gamma(K \otimes \mathcal{D}). \qquad (5.2) $$

For any $f \in C^{\infty}(M)$, we can also extend the operator $\hat{f}$ to $\text{End}_\mathbb{C}(\Gamma(K \otimes \mathcal{D}))$ as

$$ \hat{f}(s \otimes \varrho) = D_{\text{df}}(s \otimes \varrho) + 2\pi i f(s \otimes \varrho). \qquad (5.3) $$

**Proposition 5.2.** *The extended operator satisfies the prequantization condition. That is,*

$$ \widehat{\{f, g\}}(s \otimes \varrho) = \{\hat{f}, \hat{g}\}_{\varphi, \theta}(s \otimes \varrho), \quad s \otimes \varrho \in \Gamma(K \otimes \mathcal{D}). $$

*Proof.* Let $s \otimes \varrho \in \Gamma(K \otimes \mathcal{D})$. Remark that from (5.2), and (5.3) we get

$$ \hat{f}(s \otimes \varrho) = (\hat{f}(c)) \otimes \varrho + s \otimes \mathcal{L}_{\Lambda^{\#}(df)}\varrho. \qquad (5.4) $$

Then, using (5.4),(5.1), the properties of Lie derivative $\mathcal{L}$, and that of the anchor map $\Lambda^\#$, the proposition follows. $\square$

Let $\mathcal{H}_0$ be a subspace of $\Gamma(K \otimes \mathcal{D})$ defined by

$$ \mathcal{H}_0 = \{s \otimes \varrho \in \Gamma(K \otimes \mathcal{D}) \mid D_{\alpha}(s \otimes \varrho) = 0, \forall \alpha \in \mathcal{P}\}. \qquad (5.5) $$

By a Bohr-Sommerfeld type condition, we can assume that $\mathcal{H}_0 \neq \{0\}$, see [7, Page 71-72].

**Proposition 5.3.** For any $f \in \mathcal{Q}(\mathcal{P})$, and $s \otimes \varrho \in \mathcal{H}_0$, we have $\hat{f}(s \otimes \varrho) \in \mathcal{H}_0$.

*Proof.* Let $\alpha \in \mathcal{P}$, and $s \otimes \varrho \in \mathcal{H}_0$. By using (5.2), (5.3), (3.3), and the prequantization condition (4.4), we get

$$ D_{\alpha}(\hat{f}(s \otimes \varrho)) = \hat{f}(D_{\alpha}(s \otimes \varrho)) - D_{[df, \alpha]_{\varphi, \theta}}(s \otimes \varrho). $$

Then, we have

$$ D_{\alpha}(\hat{f}(s \otimes \varrho)) = \hat{f}(0) - 0 = 0. $$

Because $[df, \alpha]_{\varphi, \theta} \in \mathcal{P}$, since $f \in \mathcal{Q}(\mathcal{P})$. $\square$

Proposition 5.3 shows that the map: $\hat{f}|_{\mathcal{H}_0}: \mathcal{H}_0 \to \mathcal{H}_0$ is well-defined. Hence, $\mathcal{H}_0$ can be used as a quantization space for $\mathcal{Q}(\mathcal{P})$. To construct a Hilbert space on $\mathcal{H}_0$, we define a scalar product on $\mathcal{H}_0$. We distinguish two cases:
**Compact case** If $M$ is compact, we define

$$ \langle s_1 \otimes \varrho_1, s_2 \otimes \varrho_2 \rangle = \int_M h(s_1, s_2) \varrho_1 \bar{\varrho}_2 \qquad (5.6) $$

where $h$ is a Hermitian metric on $\xi: K \to M$, and the bar denotes the complex conjugation. This integral converges due to compactness, making $\mathcal{H}_0$ a pre-Hilbert space.

**Proposition 5.4.** The operators $\hat{f}$ for $f \in Q(\mathcal{P})$ are anti-Hermitian with respect to this scalar product.

*Proof.* Let $s_1 \otimes \varrho_1, s_2 \otimes \varrho_2 \in \Gamma(K \otimes \mathcal{D})$. According to (5.4), (5.6), and (3.2), we get

$$ \begin{aligned} \langle \hat{f}(s_1 \otimes \varrho_1), s_2 \otimes \varrho_2 \rangle + \langle s_1 \otimes \varrho_1, \hat{f}(s_2 \otimes \varrho_2) \rangle & = \langle \hat{f}(s_1) \otimes \varrho_1 + s_1 \otimes \mathcal{L}_{\Lambda^{\#}(df)} \varrho_1, s_2 \otimes \varrho_2 \rangle \\ & \quad + \langle s_1 \otimes \varrho_1, \hat{f}(s_2) \otimes \varrho_2 + s_2 \otimes \mathcal{L}_{\Lambda^{\#}(df)} \varrho_2 \rangle \\ & = \int_M \left( h(\hat{f}(s_1), s_2) + h(s_1, \hat{f}(s_2)) \right) \varrho_1 \bar{\varrho}_2 \\ & \quad + \int_M h(s_1, s_2) \left( (\mathcal{L}_{\Lambda^{\#}(df)} \varrho_1) \bar{\varrho}_2 + \varrho_1 (\mathcal{L}_{\Lambda^{\#}(df)} \bar{\varrho}_2) \right) \\ & = \int_M \Lambda^{\#}(df) \left( h(s_1, s_2) \right) \varrho_1 \bar{\varrho}_2 \\ & \quad + \int_M h(s_1, s_2) \left( (\mathcal{L}_{\Lambda^{\#}(df)} \varrho_1) \bar{\varrho}_2 + \varrho_1 (\mathcal{L}_{\Lambda^{\#}(df)} \bar{\varrho}_2) \right) \\ & = \int_M \mathcal{L}_{\Lambda^{\#}(df)} \left( h(s_1, s_2) \varrho_1 \bar{\varrho}_2 \right) = 0 \end{aligned} \qquad (5.7) $$

where equality (5.7) comes from the density version of Stokes' Theorem, see [14]. □

The Hilbert space $\mathcal{H}$ is the completion of $\mathcal{H}_0$. To obtain Hermitian operators representing quantum observables, the anti-Hermitian operators $\hat{f}$ are multiplied by the imaginary unit $i$. This transformation modifies the prequantization condition (4.3) by the same constant factor.

**Non-Compact case** If $M$ is not compact, a more involved procedure is required to define a finite scalar product. We consider the real Lie subalgebra $\mathcal{P}_0$ of $(\Omega^1(M), [,]_{\varrho,\theta})$ whose complexification is $\mathcal{P} \cap \bar{\mathcal{P}}$. We have that $\Lambda(\alpha, \beta) = 0$ for all $\alpha, \beta \in \mathcal{P}_0$. We will assume that the distribution $\Lambda^{\#}(\mathcal{P}_0)$ defines a regular foliation $\mathcal{F}$ on $M$ with a Hausdorff leaf space $N = M/\mathcal{F}$. Under these conditions, the Hamiltonian vector field $\Lambda^{\#}(df)$ of any observable $f \in Q(\mathcal{P})$ is projectable with respect to $\Lambda^{\#}(\mathcal{P}_0)$ onto $N$ (we have, $[\Lambda^{\#}(df), \Lambda^{\#}(\alpha)] = \Lambda^{\#}([df, \alpha]_{\varrho,\theta}) \in \mathcal{P}_0, \forall \alpha \in \mathcal{P}_0$). This allows us to restrict our attention to a specific type of half-densities on $M$. Specifically, we consider those that are pullbacks $\varpi^* \varrho_N$ of half-densities $\varrho_N$ on the leaf space $N$ via the canonical projection $\varpi: M \to N$. For such sections, the action of the quantum operator simplifies to:

$$ \hat{f}(s \otimes \varpi^* \varrho_N) = \hat{f}(s) \otimes \varpi^* \varrho_N + s \otimes \mathcal{L}_{\Lambda^{\#}(df)} \varpi^* \varrho_N $$
$$ = \hat{f}(s) \otimes \varpi^* \varrho_N + s \otimes \varpi^* (\mathcal{L}_{\varpi_* \Lambda^\#(df)} \varrho_N). $$

We have that $\mathcal{L}_{\Lambda^\#(\alpha)}\varpi^*\varrho_{iN} = 0$, for all $\alpha \in \mathcal{P}_0$, $i=1,2$, since $\varpi^*\varrho_{1N}, \varpi^*\varrho_{2N}$ are projectable on $N$. Using this fact, (3.2), (5.2), and (5.5), we show that the density $h(s_1, s_2)\varpi^*\varrho_{1N}\varpi^*\varrho_{2N}$ is projectable on $N$. Consequently, the scalar product of two projectable sections $s_1 \otimes \varpi^*\varrho_{1N}, s_2 \otimes \varpi^*\varrho_{2N} \in \mathcal{H}_0$ involves the projection of the quantity $h(s_1, s_2)\varpi^*\varrho_{1N}\varpi^*\varrho_{2N}$ to a 1-density $\delta_N$ on $N$. The scalar product is then given by integration over $N$:

$$ \langle s_1 \otimes \varpi^* \varrho_{1N}, s_2 \otimes \varpi^* \varrho_{2N} \rangle = \int_N \delta_N. $$

By considering the subspace $\mathcal{H}_0^c \subset \mathcal{H}_0$ of projectable sections with compact support in $N$, we obtain a pre-Hilbert space. The operators $\hat{f}$ remain anti-Hermitian with respect to this scalar product. The Hilbert space of quantum states is finally constructed by completing $\mathcal{H}_0^c$ and defining Hermitian operators as $i\hat{f}$, analogous to the compact case.

# 6 A specific quantization space on $\mathbb{R}^5$

Consider the prequantizable $\theta$-almost twisted Poisson manifold $(\mathbb{R}^5, \Lambda, \varphi, \theta)$ of Remark 4.4 with additional conditions as follows: $\Lambda = e^f \partial_{x_1} \wedge \partial_{x_2} + e^g \partial_{x_3} \wedge \partial_{x_4}$, $\varphi = e^{-g} dx_3 \wedge dx_4 \wedge dx_5 + e^{-f} dx_1 \wedge dx_2 \wedge dx_5$, and $\theta = dx_5$ where $f = f(x_1, x_2)$, and $g = g(x_3, x_4)$. As we have seen, the solution of (4.7) is $(Z, \eta)$ where $Z = \partial_{x_5}$, and $\eta = -e^{-f} \partial_{x_1} \wedge \partial_{x_2} - e^{-g} \partial_{x_3} \wedge \partial_{x_4}$.

The prequantization bundle of $(\mathbb{R}^5, \Lambda, \varphi, \theta)$ is the trivial complex line Bundle $K = \mathbb{R}^5 \times \mathbb{C}$ with the usual Hermitian metric $h$ and the contravariant derivative $D$ defined by

$$ D_{\alpha}s = \Lambda^{\#}(\alpha)s, \qquad (6.1) $$

for all $\alpha \in \Omega^1(\mathbb{R}^5)$, $s \in \Gamma(\mathbb{R}^5 \times \mathbb{C}) = C^\infty(\mathbb{R}^5, \mathbb{C})$. We can identify $\mathbb{R}^5$ to $\mathbb{C}^2 \times \mathbb{R}$. Then, we set: $z_k = x_{2k-1} + ix_{2k}$ and $\bar{z}_k = x_{2k-1} - ix_{2k}$ with $i = 1, 2$. This gives $dx_{2k} = \frac{1}{2}(dz_k + d\bar{z}_k)$, $dx_{2k-1} = -\frac{i}{2}(dz_k - d\bar{z}_k)$, $\partial_{x_{2k-1}} = \partial_{z_k} + \partial_{\bar{z}_k}$, and $\partial_{x_{2k}} = i(\partial_{z_k} - \partial_{\bar{z}_k})$. Thus, in the complex coordinates $(z_1, z_2, t)$ of $\mathbb{R}^5$, the pair $(\Lambda, \varphi)$ is written as follow:

$$ \Lambda = -2i[e^f \partial_{z_1} \wedge \partial_{\bar{z}_1} + e^g \partial_{z_2} \wedge \partial_{\bar{z}_2}], $$

$$ \varphi = -\frac{i}{2}[e^{-f} dz_1 \wedge d\bar{z}_1 + e^{-g} dz_2 \wedge d\bar{z}_2] \wedge dt. $$

A convenient complex polarization of $(\mathbb{R}^5, \Lambda, \varphi, \theta)$ is $\mathcal{P} = \text{span}\{dz_1, dz_2\}$, and we see that $\mathcal{P}(\mathcal{P})$ consists of the functions $h \in C^\infty(\mathbb{R}^5)$ such that

$$ [dh, dz_k]_{\varphi, \theta} \in \mathcal{P}, \quad i = 1, 2. $$

After a computation, $[dh, dz_k]_{\varphi, \theta} \in \mathcal{P}$ if and only if,

$$ \partial_{\bar{z}_k} h = 0, \quad k = 1, 2. \qquad (6.2) $$
We consider $\widetilde{P(\mathcal{P})}$ the set of pair $(h_1, h_2)$ of solutions of (6.2) for that

$$[i_{\Lambda^{\#}(dh_2)} i_{\Lambda^{\#}(dh_1)} \varphi + \Lambda(dh_1, dh_2) dt, dz_k]_{\varphi, \theta} \in \mathcal{P}, \quad k = 1, 2.$$

We take $\mathcal{Q}(P)$ to be the projection of $\widetilde{P(\mathcal{P})}$ on $P(\mathcal{P})$. Remark that $t \in \mathcal{Q}(P)$.

To determine the quantization space $\mathcal{H}_0$, we need the bundle of complex half-densities $\mathcal{D}$ over $\mathbb{R}^5 = \mathbb{C}^2 \times \mathbb{R}$. A basis of $\mathcal{D}$ can be written as $\beta = |v|^{\frac{1}{2}}$ where

$$v = dx_1 \wedge \cdots \wedge dx_5 = -\frac{1}{4} dz_1 \wedge dz_2 \wedge d\bar{z}_1 \wedge d\bar{z}_2 \wedge dt.$$

If we take 1 as a basis of $K = \mathbb{R}^5 \times \mathbb{C}$, then any section $s \otimes \varrho$ of $K \otimes \mathcal{D}$ can be written as $s \otimes \varrho = 1 \otimes (\chi\beta)$, where $\chi \in C^{\infty}(\mathbb{R}^5, \mathbb{C})$.

If we denote by $\mathcal{D}$ the extension (5.2) of the Hermitian contravariant derivative $\mathcal{D}$ given by (6.1), we obtain

$$D_{dz_k}(1 \otimes (\chi\beta)) = 1 \otimes \mathcal{L}_{\Lambda^{\#}(dz_k)}(\chi\beta), \quad k = 1, 2. \qquad (6.3)$$

Therefore, Using the fact that $\mathcal{L}_X\gamma = (\frac{1}{2} \mathrm{div} X)\gamma$ (see, for instance,[8]), it follows that

$$D_{dz_k}(1 \otimes (\chi\beta)) = 0$$

if and only if

$$\mathcal{L}_{\Lambda^{\#}(dz_k)}(\chi)\beta + \frac{\chi}{2} \mathrm{div} \Lambda^{\#}(dz_k)\beta = 0. \qquad (6.4)$$

But $\Lambda^{\#}(dz_k) = -2ie^{A_k} \partial_{\bar{z}_k}$ and $\mathrm{div} \Lambda^{\#}(dz_k) = -2ie^{A_k} \partial_{\bar{z}_k} A_k$, with $A_1 = f, A_2 = g$. Hence, equation (6.1) is equivalent to

$$\partial_{\bar{z}_k} \chi + \frac{\chi}{2} \partial_{\bar{z}_k} A_k = 0.$$

Thus the quantization space $\mathcal{H}_0$ can be identified with the space

$$\mathcal{H}_0 = \{ \chi \in C^{\infty}(\mathbb{R}^5, \mathbb{C}) \mid \partial_{\bar{z}_k} \chi + \frac{\chi}{2} \partial_{\bar{z}_k} A_k = 0, \forall k = 1, 2 \}.$$

Note that $e^{-\frac{1}{2}f+t}$, and $e^{-\frac{1}{2}g+t}$ are elements of $\mathcal{H}_0$. This means $\mathcal{H}_0$ is non-trivial. Let $h \in \mathcal{Q}(P)$, and $\chi \in \mathcal{H}_0$. Using (4.1),(5.3), we deduce that

$$\hat{h}(\chi) = 2\pi i h \chi + \Lambda(dh, d\chi) + \frac{\chi}{2} \mathrm{div} \Lambda^{\#}(dh).$$

Moreover, the scalar product of two function $\chi_1, \chi_2 \in \mathcal{H}_0$ with compact support is

$$\langle \chi_1, \chi_2 \rangle = \int_{\mathbb{R}^5} \chi_1 \overline{\chi_2} \beta.$$

## References

[1] A. Chatzistavrakidis and G. Šimunic, *Gauged Sigma Models with nonclosed 3-Form and twisted Jacobi structure*. J.High Energ. Phys.2020. 173(2020).

[2] A. Nijenhuis, *Jacobi-type identities for bilinear differential concomitants of certain tensor fields*, I, II, Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math. 17 (1955), 390-397, 398-403.
[3] B. Kostant, *Quantization and Unitary Representations*, Springer Berlin Heidelberg. pp.87-208,1970.

[4] F. Petalidou, *On the geometric quantization of twisted Poisson manifolds*, J. Math. Phys. 48 (8), 2007.

[5] J. A. Schouten, *On the differential operators of first order in tensor calculus*, Convegno Internazionale di Geometria Differenziale, Italia, 1953, Edizioni Cremonese, Roma, 1954, pp. 1-7.

[6] J. Huebschmann, *Poisson cohomology and quantization*, j. reine angew. Math. 408, 57-113, 1990. 271.

[7] J. Śniatycki, *Geometric Quantization and Quantum Mechanics* (Springer, Berlin, 1980).

[8] K. Yano, *The theory of Lie derivatives and its applications*, North Holland Publ., Amsterdam, 1957.

[9] M. Crainic, R. L. Fernandes, I. Marcut, *Lectures on Poisson Geometry, Graduate Studies in Mathematics*, Graduate Studies in mathematics 217, Amer. Math. Soc, 2021.

[10] N. Saipele Nansidi, B. Tangue, J. Dongho, *θ-almost twisted Poisson cohomology*, arXiv:2502.18503 [math.DG], 2025.

[11] P. Ševera and A. Weinstein, *Poisson geometry with a 3-form background*, Non-commutative geometry and string theory, Prog. Theoret. Phys. Suppl. 144, 2001, 145-154.

[12] S. Bates, A. Weinstein, *Lectures on the Geometry of Quantization* (A.M.S., Berkeley Mathematics Lecture Notes 8, Amer. Math. Soc., 1997).

[13] S. Kobayashi, K. Nomizu, *Foundations of Differential Geometry*, volume 1, Wiley-Interscience, New York, 1969.

[14] S. Sternberg, *Lectures on Differential Geometry* (Prentice-Hall, Englewood Cliffs, 1964).