Deformation Quantization of Prequantizable Fibrations

• The paper establishes a rigorous method for constructing associative quantum algebras and modules for fibered symplectic manifolds using strict deformation quantization.
• It employs real polarization and Toeplitz operator calculus to ensure operator convergence and accurately reproduce classical Poisson brackets via star products.
• The approach bridges geometric quantization, microlocal analysis, and index theory, providing a coherent framework for quantizing families of integrable systems.

Deformation quantization of prequantizable symplectic fibrations is a program for constructing associative quantum algebras and modules for classical phase spaces organized as symplectic manifolds fibered over a base, with each fiber individually admitting geometric quantization. This approach systematically generalizes geometric and deformation quantization to bundles whose fibers carry compatible prequantum structures. The resulting algebras and Hilbert bundles supply a rigorous mathematical realization of quantum mechanics for families of integrable or almost integrable systems, and connect microlocal analysis, operator algebras, geometric quantization, and modern index theory.

1. Geometric Structure and Prequantizability

Let $M^{2n}$ be a symplectic manifold with nondegenerate closed 2-form $\omega$. A proper Lagrangian fibration is a smooth surjection $\pi: M^{2n} \to B^n$ whose fibers $X_b = \pi^{-1}(b)$ are compact, connected, and Lagrangian ($\omega|_{X_b}=0$), and such that $\pi$ is a locally trivial fibration. A symplectic fibration $p: S \to B$ is a fiber bundle with closed 2-form $\Omega$ restricting as a symplectic form on each fiber $S_b$. Prequantizability requires the integrality condition $[\omega/2\pi] \in H^2(M;\mathbb{Z})$ (resp. $[\Omega/2\pi] \in H^2(S;\mathbb{Z})$) so that there exists a Hermitian line bundle $(L,h)$ over $M$ (resp. $S$) with unitary connection $\nabla$ and curvature $\mathrm{curv}(\nabla) = -i\omega$ (resp. $-i\Omega$) (Yamashita, 2020, Duval et al., 2011, Cren et al., 10 Jan 2026).

Equivalently, there is a principal $U(1)$ or $S^1$ bundle $Y \to M$ with connection 1-form $\alpha$ satisfying $d\alpha = \omega/\hbar$ for a (possibly fixed) Planck parameter $\hbar$; $M$ (resp. each fiber) is prequantizable in this sense. In the case of families, such line bundles and connections glue to smooth bundles over the total space, with connections inducing fiberwise symplectic structures (Duval et al., 2011).

2. Quantum Hilbert Spaces and Real Polarization

Associated to a prequantizable Lagrangian fibration, one constructs quantum Hilbert spaces via real polarization. For each integer $k\ge 1$, the set $B_k\subset B$ of $k$-Bohr–Sommerfeld points is defined by $H^0(X_b;L^k)\ne \{0\}$. The quantum space at level $k$ is

$$H_k = \bigoplus_{b\in B_k} H^0\left(X_b, L^k\otimes |\Lambda|^{1/2}X_b\right)$$

where $|\Lambda|^{1/2}X_b$ denotes the vertical half-density bundle, canonically flat. Each summand is one-dimensional; in action–angle coordinates one finds $B_k \simeq \tfrac{1}{k}\mathbb{Z}^n$ and canonical Fourier bases $$\psi_b^k(\theta) = e^{ik\langle b, \theta\rangle}\sqrt{d'\theta}$$ for $b\in B_k$ (Yamashita, 2020).

When considering families, one obtains a Hilbert bundle over $B$ with fiberwise quantum spaces, and a smooth field of quantum states and operators.

3. Strict Deformation Quantization: Operator Constructions

The key to deformation quantization in this setting is the construction of a strict deformation quantization—an assignment $f\mapsto Q_k(f)$ of bounded operators $Q_k(f): H_k\to H_k$ for each $f\in C^\infty(M)$, such that operator commutators asymptotically reproduce the Poisson bracket:

$$\left[Q_k(f),Q_k(g)\right] + \frac{i}{k}Q_k(\{f,g\}) = O(k^{-2})$$

and $\|Q_k(f)\|\to \|f\|_{C^0}$ as $k\to\infty$ (operator norm convergence to the supremum norm) (Yamashita, 2020).

In local models such as $\mathbb{R}^n\times T^n$, $Q_k(f)$ acts by a lattice Fourier kernel:

$$K_f(b,c) = \int_{T^n} e^{-ik\langle b-c,\theta\rangle} f\left(\frac{b+c}{2},\theta\right)d'\theta$$

and in general, parallel transport and localization control the matrix elements, using a horizontal distribution on $M$. The construction is a "lattice approximation" to the principal symbol-operator correspondence.

The approach is compatible with the real-polarization quantum spaces, and off-diagonal terms in the kernel decay sufficiently to guarantee strict control over operator asymptotics.

4. Formal Star Products and Comparison with Fedosov/Berezin–Toeplitz

From the strict family $\{Q_k\}$, one extracts a formal star product on $C^\infty(M)[[\hbar]]$ by the asymptotic expansion

$$Q_k(f) Q_k(g) = \sum_{j=0}^l \left(\frac{-i}{k}\right)^j Q_k(C_j(f,g)) + O(k^{-l-1}), \quad \hbar = 1/k$$

yielding

$$f\star g = \sum_{j=0}^\infty \hbar^j C_j(f,g), \qquad C_0(f,g)=fg, \quad C_1(f,g)=\frac{i}{2}\{f,g\}$$

and higher $C_j$ bidifferential operators of order $j$ in each argument. In the standard torus model, this reproduces the Moyal–Weyl product. For general torus bundles or Abelian varieties, the resulting product agrees to order $\hbar^2$ with the Fedosov star product constructed from the canonical torsion-free symplectic connection determined by the chosen horizontal lift (Yamashita, 2020). In the presence of a compatible complex structure (Kähler), the quantization overlaps with the Berezin–Toeplitz construction: $Q_k(f)$ and Berezin–Toeplitz operators $T^k(f)$ converge in norm as $k\to\infty$ (Yamashita, 2020).

Toeplitz operator calculus further allows an analytic approach: using smooth families of Szegö projectors on the fibers and S$^1$-equivariance, one builds Toeplitz operator families whose composition yields the same bidifferential algebra and star product as in algebraic (Fedosov-type) deformation quantization. The first bidifferential terms are universal: $C_0(f,g) = fg$, $C_1(f,g) = \{f,g\}$, $C_2$ includes covariant derivatives and curvature terms (Cren et al., 10 Jan 2026).

Construction	Fiberwise Module	Operator Realization	Star Product
Lattice-Fourier	Yes	$Q_k(f)$	Fedosov (up to $\hbar^2$)
Toeplitz families	Yes	Toeplitz, Szegö	Equivalent to Fedosov
Deformation–Prequant	Yes	Bidifferential in $Y$	Fiberwise, module

5. Deformation of Prequantization and Module Structures

Deformation quantization can proceed not only at the algebra of observables, but directly via modules on prequantum bundles. Given a principal $U(1)$-bundle $Y\to X$ (or $Y\to S$ in fibrations) with connection 1-form $\alpha$ ($d\alpha = \omega/\hbar$) and horizontal distribution, the horizontal lift of the Poisson bivector, called the Souriau bracket $\{f,g\}_S$, defines a lift of the Poisson structure to functions on $Y$:

$\{f,g\}_S = \pi^\#(df,dg)$

where $\pi^\#$ is the $U(1)$-invariant horizontal lift. On $U(1)$-equivariant functions (the "prequantum wave functions"), one constructs an associative quantum product:

$$f\bullet g = fg + \frac{\hbar}{i}\{f,g\}_S + \cdots$$

In fact,

$$f\bullet g = m\circ \exp\left(\frac{\hbar}{i}\pi^\#\right)(f\otimes g)$$

The result is a module structure over the star algebra of $C^\infty(X)[[\hbar]]$, compatible with chosen polarizations and producing Hilbert bundles whose fibers furnish quantizations of the symplectic fibers. For prequantizable symplectic fibrations, the construction localizes and extends fiberwise, and in the case of a product fibration (e.g., $M\times T^*\mathbb{R}^n$), yields the standard quantum operator representations with normal ordering on each fiber (Duval et al., 2011).

6. Index Theory and Topological Invariants

The deformation quantization of symplectic fibrations supports a natural index theory. Families of Toeplitz operators on contact fibrations (which are $S^1$-bundles over the symplectic base) yield K-theory classes in $KK_0(X,B)$. The Fredholm index bundle is given by the push-forward of the Chern character and vertical Todd class:

$$\operatorname{Ind}(T_f) = \pi_{X!}\left(\mathrm{ch}(f)\smile\mathrm{Td}(T(X/B))\right)\in H^{\mathrm{even}}(B;\mathbb{Q})$$

where $T(X/B)$ is the vertical tangent bundle for the fibration $X\to B$, and $\int_{X/B}$ denotes fiber integration. For scalar Toeplitz families, this generalizes classical fiberwise index formulas of Boutet de Monvel to a family setting, connecting geometric quantization, microlocal analysis, and topological K-theory invariants (Cren et al., 10 Jan 2026).

7. Context, Comparisons, and Applications

The analytic Toeplitz approach and the algebraic Fedosov construction yield, up to equivalence, the same star product and deformation class for prequantizable symplectic fibrations. The analytic (operator-theoretic) methods use spectral data and the Heisenberg calculus, while algebraic methods use formal geometry and characteristic classes (Cren et al., 10 Jan 2026).

Deformation quantization via prequantum circle bundles offers an explicit module viewpoint, allowing construction fiberwise across the fibration and adapting directly to chosen polarizations (Duval et al., 2011). The compatibility of these constructions underlies the geometric quantization of families of integrable systems, their representation spaces, and topological invariants. Applications include the quantization of Lagrangian torus fibrations, Hilbert bundles over moduli spaces of polarizations, and explicit comparison of quantization methods in Kähler and non-Kähler examples (Yamashita, 2020, Duval et al., 2011, Cren et al., 10 Jan 2026).

Necessary hypotheses are: properness (compactness) of fibers, integrality of the symplectic class, regularity in the sense of Arnold–Liouville, and a choice of horizontal distribution. Under these, strict $C^*$-algebraic deformation quantization is achieved, with quantum representation spaces compatible with geometric quantization.