Geometric Quantization Overview

• Geometric Quantization is a method that converts classical phase space data into quantum Hilbert spaces through symplectic geometry and prequantum line bundles.
• It employs techniques such as prequantization, polarization (real, complex, and half-form), and sheaf cohomology to construct physically meaningful operator representations.
• Recent advances include polarization-free approaches via topological field theory and extensions to generalized settings like Poisson and Dirac manifolds.

Geometric quantization is a rigorous procedure that constructs quantum Hilbert spaces and operator representations from the differential-geometric data of classical phase spaces, typically symplectic manifolds. Central to the formalism is the passage from smooth functions and symplectic geometry to unitary representations on Hilbert spaces and noncommutative operator algebras, subject to integrality, polarization, and potential metaplectic corrections. Recent advances include polarization-free constructions using topological field theory, analytic classification of Hilbert fields, and generalized quantization for Poisson, Dirac, presymplectic, and b-symplectic manifolds.

1. Symplectic and Poisson Manifolds: Foundations

A symplectic manifold $(M^{2n},\omega)$ consists of a smooth, even-dimensional manifold equipped with a closed, nondegenerate 2-form $\omega \in \Omega^2(M)$, $d\omega = 0$, with $\omega_p:T_pM \times T_pM \to \mathbb{R}$ invertible for all $p$ (Berktav et al., 2 Dec 2025). The associated Poisson bracket for $f,g \in C^\infty(M)$ is

$\{f,g\} = \omega(X_g,X_f)$

where $X_f$ is defined by $\iota_{X_f}\omega = df$. Symplectic actions by Lie groups $G$ admit moment maps $\mu:M \to \mathfrak{g}^*$ characterized by $d\langle \mu,X \rangle = \iota_{X^M}\omega$ for all $X \in \mathfrak{g}$.

Geometric quantization generalizes to Poisson manifolds $(M,\Pi)$ and Dirac manifolds $(M,D)$, with the Poisson sigma model and Lie algebroid cohomology supplying the requisite algebraic structure and admissible function spaces (Hirota, 2013).

2. Prequantization: Line Bundles and Integrality

A prequantum line bundle $L \to M$ is a Hermitian complex line bundle with unitary connection $\nabla$ and curvature

$F_\nabla = \nabla^2 = -\frac{i}{\hbar}\omega$

where the integrality condition $[\omega]/(2\pi \hbar) \in H^2(M;\mathbb{Z})$ is necessary and sufficient for existence (Berktav et al., 2 Dec 2025, Lackman, 2024). Locally, given $\omega = dA$, one has $\nabla = d - (i/\hbar)A$, with transition functions $e^{-(i/\hbar)f_{ij}}$. These bundles are unique up to tensoring with flat bundles, classified via differential cocycle stacks (Lerman, 2012).

For Dirac manifolds, prequantization utilizes line bundles $L \to M$ with $D$-connection whose curvature $R^D = 2\pi i A$, where $A$ is the relevant Lie algebroid 2-cocycle (Hirota, 2013). In presymplectic geometry, only the pull-back of the integral cohomology under the anchor map counts.

3. Quantization Map, Polarizations, and Half-Form Corrections

The canonical prequantization operator for $f \in C^\infty(M)$ is

$Q(f) = -i\hbar\,\nabla_{X_f} + f$

acting on square-integrable sections $s \in L^2(M, L)$ (Berktav et al., 2 Dec 2025). The Dirac commutator obeys

$[Q(f), Q(g)] = -i\hbar Q(\{f,g\})$

at the prequantum level.

Polarization is required to select the physical Hilbert space:

• Real polarization: Integrable Lagrangian distributions $P \subset TM$ of rank $n$ restrict quantization to covariantly constant sections along $P$ (Schrödinger representation) (Lackman, 2024, Berktav et al., 2 Dec 2025, Miranda et al., 2013).
• Kähler (complex) polarization: $P = T^{0,1}M$ yields holomorphic sections (Bargmann–Fock representation) (Berktav et al., 2 Dec 2025).
• Metaplectic (half-form) correction: For well-posed inner products and correct spectral shifts (e.g. harmonic oscillator $+\frac{1}{2}\hbar$), one twists by the square root of the canonical bundle, forming $L \otimes \sqrt{K_P}$ or half-densities $L \otimes \delta_P^{1/2}$ (Berktav et al., 2 Dec 2025, Lerman, 2012).

Flat sections of the corrected bundle along $P$ define cohomological quantization spaces, computed via sheaf cohomology (Kostant complex), which respects Mayer–Vietoris and Künneth formulae (Miranda et al., 2013, Mir et al., 2021).

4. Polarization Independence and the Role of the Poisson Sigma Model

Traditional geometric quantization depends on the choice of polarization, leading to ambiguities and challenging uniqueness problems. The path-integral approach of the Poisson sigma model bypasses the need for polarization by constructing a canonical quantization map via topological field theory (Lackman, 2024):

• The quantization map $Q: C^\infty(M) \to B(\mathcal{H})$ is defined as a convolution algebra of multiplicative line bundles over the 2-groupoid of $M$ based on disk and sphere integrals,
• The semi-classical expansion yields the Kontsevich star-product,
• The construction descends to the source-simply-connected symplectic groupoid and recovers operator algebras of interest (e.g. noncommutative torus).

Crucially, by acting on the full prequantum Hilbert space, the construction allows Schur’s lemma to guarantee that different polarizations (when their spaces overlap) are canonically (projectively) unitarily isomorphic. The Blattner–Kostant–Sternberg (BKS) pairing forms the technical backbone of this identification.

5. Sheaf Cohomology: Quantization of Real Polarizations

For real polarizations associated to Lagrangian fibrations (e.g. toric, action–angle coordinate systems), quantization spaces are computed as the cohomology $H^j(M, \mathcal{J})$ of sheaves of flat sections of the prequantum bundle, possibly with half-form correction (Miranda et al., 2013).

• The space is supported on Bohr–Sommerfeld leaves, i.e. those on which the holonomy of the prequantum bundle is trivial,
• For fibrations over simply connected bases, the quantization dimension equals the number of Bohr–Sommerfeld fibers (Sniatycki's theorem),
• The Kostant complex and leaf-wise de Rham cohomology supply Mayer–Vietoris and Künneth tools for recursive calculation,
• Generic regular torus foliations and irrational flows admit computation of quantization spaces and capture phenomena such as infinite-dimensional $H^1$ for Liouville irrationality,
• In integrable systems with non-degenerate singularities (elliptic, hyperbolic, focus–focus), only analytically flat sections survive for hyperbolic singularities; focus–focus blocks contribute as regular tori via cylinder surgery (Mir et al., 2021).

6. Polarization Independence: Analytic Fields, SYZ Transforms, and Families

Geometric quantization often produces entire families of Hilbert spaces parametrized by auxiliary geometric data (complex structures $J$, Kähler structures, etc.). Analytic Hilbert fields admit smooth or analytic sections and connections; flatness of the connection implies the existence of a canonical trivialization and path-independent parallel transport (Lempert et al., 2010).

• For compact homogeneous spaces, quantization can be flat (uniquely determined up to unitary isomorphism),
• For Kähler polarizations, metric and half-form corrections can be analyzed using Toeplitz factors and curvature computations,
• SYZ transforms solve the polarization-independence problem for semi-flat Lagrangian torus fibrations and smooth projective toric manifolds by constructing canonical isomorphisms between real and complex polarized spaces, using Fourier-theoretic and Morse-theoretic correspondences between intersection points of Lagrangians (on the A-side) and holomorphic sections (on the B-side) (Chan et al., 2018).

7. Applications: Noncommutative Torus, Moduli Spaces, Reduction Principles

The methodology recovers classical and modern quantization instances:

• Noncommutative torus $T^2$: Quantization via disk integrals produces the standard Moyal star-product and finite-dimensional representations on holomorphic (theta-function) sections, with commutation relations $UV = e^{2\pi i\theta}VU$ (Lackman, 2024),
• Moduli of flat connections (Chern–Simons theory): Geometric quantization via Kähler polarization and determinant line bundles yields spaces of holomorphic sections whose dimension is given by the Verlinde formula; knot invariants are encoded in expectation values of Wilson loops (Berktav et al., 2 Dec 2025, Dey, 2016),
• Quantization commutes with reduction: For Hamiltonian Lie group actions, the index of the Dirac operator on the original symplectic manifold matches that of the reduced space, even in noncompact settings provided technical controls on cocompactness and the spectrum of deformation vector fields (Braverman et al., 2019, Hochs et al., 2013, Lin et al., 2022),
• Generalizations: Quantization extends to shifted symplectic stacks, b-symplectic manifolds (with APS index), presymplectic manifolds, and Dirac structures, with methods adapted to leafwise cohomology, topological obstructions, and representation-theoretic data (Safronov, 2020, Braverman et al., 2019, Hirota, 2013, Lin et al., 2022).

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Table: Classical Quantization Steps and Key Objects

Step	Classical Data	Quantized Object
Symplectic manifold	$(M, \omega)$	—
Prequantization	Line bundle $L \to M$, $\nabla$	Hilbert space $L^2(M, L)$
Polarization (real/complex)	Involutive Lagrangian $P$	Polarized sections
Metaplectic correction	Half-form bundle $\delta_P$	$L \otimes \delta_P$
Operator assignment	$f \mapsto Q(f)$	Quantum operator $Q(f)$
Invariance of polarization	—	BKS pairing, Schur’s lemma
Path-based/polarization-free	Poisson sigma model	$Q$ via convolution algebra

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Geometric quantization thus provides a universal, functorial, and cohomological approach to the passage from classical to quantum mechanics, extending naturally to complex and singular geometries, equivariant and derived settings, and underpinning rigorous constructions in representation theory and topological field theory. Recent advances in polarization-free quantization, analytic Hilbert field classification, and rigorous sheaf-theoretic approaches have resolved key ambiguities and expanded the formal requirement for quantization, ensuring canonical identifications across polarizations and extending the method to broad classes of manifolds and stacks (Lackman, 2024, Chan et al., 2018, Lempert et al., 2010, Miranda et al., 2013).