Prequantization Bundle Overview

• Prequantization bundles are principal S¹-bundles with connections whose curvature pulls back the symplectic form, satisfying the integrality condition for quantization.
• They induce a canonical contact structure with a Reeb vector field generating periodic orbits, enabling rigorous analysis via Floer and capacity theories.
• Applications span geometric quantization, moduli space topology, and higher structures, linking analytic, topological, and algebraic methods in modern symplectic geometry.

A prequantization bundle is a geometric structure providing the foundational data for geometric quantization of symplectic manifolds. It is canonically constructed as a principal $S^1$-bundle (or $U(1)$-bundle) endowed with a connection whose curvature reproduces the given symplectic form, yielding an associated contact manifold with a distinguished Reeb dynamics. Prequantization bundles encode the integrality condition of the symplectic class and play central roles in symplectic topology, contact geometry, quantization theory, and the topology of moduli spaces.

1. Construction and Fundamental Properties

Given a closed symplectic manifold $(M,\omega)$ with $[\omega] \in H^2(M; \mathbb{R})$ admitting an integral lift $[\omega] \in H^2(M; \mathbb{Z})$, a prequantization bundle is a principal $S^1$-bundle $\pi: P \to M$ whose Euler class (or first Chern class) satisfies $c_1(P) = [\omega]/2\pi$. Such a bundle admits an $S^1$-invariant connection form $\alpha \in \Omega^1(P)$ with curvature $d\alpha = \pi^*\omega$ (Vaughan, 2014).

The existence of the prequantization bundle is equivalent to the integrality of $[\omega]/2\pi$—this is the quantization condition from geometric quantization (Vaughan, 2014, Albers et al., 2016). The connection form $\alpha$ defines a contact structure $\xi = \ker \alpha$ on $P$, with the Reeb vector field generating the $S^1$-action, and $P$ is then also a contact manifold (Opshtein, 21 Dec 2025).

Key properties:

• The total space $P$ of the bundle is a contact manifold, the fibers of $\pi$ are the Reeb orbits of $\alpha$.
• Locally, in a trivialization, $\alpha = d\theta + A$, where $\theta$ is the angular coordinate and $dA = \omega$.
• The construction generalizes to higher rank and infinite-dimensional settings using equivariant and bundle gerbe perspectives (Krepski, 2014, Perez, 2017, Bunk et al., 2016).

2. Contact Geometry and Reeb Dynamics

The canonical contact form $\alpha$ on the total space $P$ makes $(P,\alpha)$ a contact manifold of Boothby–Wang type (Albers et al., 2016, Opshtein, 21 Dec 2025). The Reeb vector field $R_\alpha$ is characterized by $\iota_{R_\alpha}d\alpha=0$ and $\alpha(R_\alpha)=1$, serving as the generator of the $S^1$-action.

Dynamically, all fibers are periodic Reeb orbits, and more generally, perturbations of $\alpha$ yield rich nondegenerate Reeb dynamics. In particular, Hamiltonian and Floer-theoretic techniques allow for detailed analysis of periodic Reeb orbits:

• Existence results yield lower bounds on the multiplicity of simple closed Reeb orbits in terms of $\cuplength(M)$ for graphical hypersurfaces in prequantization bundles (Albers et al., 2016).
• The nature of Reeb dynamics under these structures is crucial in the study of symplectic and contact invariants such as contact homology, embedded contact homology (ECH), and Rabinowitz Floer homology (Nelson et al., 2020, Bae et al., 2023, Ginzburg et al., 2018).

3. Symplectic Fillings and Holomorphic Foliations

A strong symplectic filling of a prequantization bundle $(V, \alpha)$ is a symplectic manifold $(W, \Omega)$ with $\partial W = V$ such that near the boundary, $\Omega(Y, \cdot)|_V = \alpha$ for a Liouville vector field $Y$ pointing outward (Chen, 2024). Asphericity conditions, capacity finiteness, and topological constraints on $W$ lead to significant classification results:

• If $W$ is symplectically aspherical and certain topological and capacity conditions are met, $W$ is diffeomorphic to the disk bundle associated to the underlying complex line bundle (Chen, 2024).
• J-holomorphic curve theory, specifically the existence and uniqueness of embedded holomorphic planes asymptotic to Reeb fibers, leads to a holomorphic foliation of $W$ by disks, producing a smooth bundle map $W \to M$ identifying $W$ as the disk bundle (Chen, 2024).
• These results connect analytic, topological, and global symplectic geometry methods, utilizing Siegel/Gutt–Hutchings capacities and intersection theory.

4. Topology, Homology, and Invariants

Prequantization bundles encode deep topological data:

• The homology of $P$ reflects both the topology of the base and the nature of the circle action; e.g., for base $\Sigma_g$ of genus $g$, $H_1(P)\cong H_1(\Sigma_g) \oplus \mathbb{Z}/(-e)$ (Nelson et al., 2020).
• ECH of contact prequantization bundles over Riemann surfaces identifies with the exterior algebra on homology of the base, precisely as graded vector spaces, exhibiting stability and structural isomorphism with Seiberg–Witten Floer theory (Nelson et al., 2020).

Symplectic homology, filtered by linking number or action, provides algebraic structures tied to the prequantization geometry. For aspherical bases, equivariant symplectic homology decomposes according to the free homotopy class of closed Reeb orbits, yielding new proofs of results such as the Conley conjecture: the existence of infinitely many simple closed Reeb orbits (Ginzburg et al., 2018).

5. Prequantization in Quantization Theory

From the perspective of geometric quantization, the prequantization bundle serves as the first step in quantizing a symplectic manifold:

• The Kostant–Souriau construction associates to $(M,\omega)$ a prequantum line bundle (Hermitian line bundle with compatible connection), with curvature $F_\nabla=-2\pi i\,\omega$ (Vaughan, 2014).
• The Lie algebra of infinitesimal quantomorphisms (vector fields on $P$ preserving the connection form) is canonically isomorphic to the Poisson algebra $C^\infty(M)$ (Vaughan, 2014).
• The bundle can be extended to more general settings, including metaplectic-c structures and polysymplectic manifolds, to accommodate second quantization and field-theoretic constructions (Vaughan, 2014, Blacker, 2019).

Prequantization bundles enable the passage from classical to quantum observables by ensuring the integrality needed for the existence of global quantum line bundles.

6. Applications and Extensions

Prequantization bundles are central in diverse mathematical contexts:

• Contact Non-Squeezing and Legendrian Barriers: Explicit Legendrian submanifolds in prequantization bundles serve as universal interlinkers obstructing certain contact embeddings, with applications to the contact non-squeezing problem (Opshtein, 21 Dec 2025).
• Floer Theoretic Correspondences: The Fukaya category of non-exact rational Lagrangians in an integral symplectic manifold can be computed via exact Lagrangians in a filling of the prequantization bundle, establishing Fukaya-sheaf correspondences (Kuwagaki et al., 2024).
• Moduli Spaces and Equivariant Prequantization: Prequantum bundles generalize to infinite-dimensional and moduli spaces, such as those for flat bundles and connections, using equivariant Chern–Simons theory and differential characters (Perez, 2017). In the context of surfaces, this includes prequantum bundles for the Weil–Petersson form on Teichmüller space or for moduli spaces of flat bundles (Krepski, 2014).
• 2-Plectic and Higher Analogues: The theory extends to 2-plectic manifolds, where prequantization proceeds via bundle gerbes and their Dixmier–Douady classes, yielding categorified analogues and 2-Hilbert spaces (Sevestre et al., 2020, Bunk et al., 2016).

7. Generalizations and Higher Structures

Classical prequantization bundles are further extended in several directions:

• Metaplectic-c and Higher Quantization: Metaplectic-c structures and associated prequantization bundles broaden the class of quantizable symplectic manifolds and refine the space of quantomorphisms (Vaughan, 2014).
• Polysymplectic Prequantization: For vector-valued forms, prequantum vector bundles admit actions of the space of coefficients, and curvature is required to reproduce polysymplectic forms (Blacker, 2019).
• Prequantization Gerbes: On 2-plectic manifolds, bundle gerbes with connection and curving capture the data of closed 3-forms, with quantization functorially yielding strict 2-Hilbert spaces of sections (Sevestre et al., 2020, Bunk et al., 2016).

Summary Table: Core Data of a Prequantization Bundle

Structure	Definition / Property	Reference
Principal $S^1$-bundle	$\pi: P \to M$, $c_1(P) = [\omega]/2\pi$	(Vaughan, 2014)
Connection 1-form	$\alpha$, $d\alpha = \pi^*\omega$	(Albers et al., 2016)
Contact manifold	$(P, \alpha)$, $\ker\alpha$ contact structure	(Opshtein, 21 Dec 2025)
Curvature integrality	$[\omega]/2\pi \in H^2(M; \mathbb{Z})$	(Vaughan, 2014)
Reeb vector field	Generator of $S^1$-action, periodic orbits = fibers	(Albers et al., 2016)
Quantomorphisms	Lie algebra isomorphic to $C^\infty(M)$ (Poisson)	(Vaughan, 2014)
Holomorphic foliation	Filling by holomorphic disks under capacity constraints	(Chen, 2024)
Rabinowitz/Floer theory	Exact sequences and computation via prequantization	(Bae et al., 2023)
2-plectic generalization	Gerbe with Dixmier–Douady class, 2-Hilbert space	(Bunk et al., 2016)

Prequantization bundles thus provide the essential geometric infrastructure bridging symplectic geometry, contact topology, and quantization, serving as a locus for the interplay of topological, analytic, and algebraic structures in modern mathematical physics and symplectic topology.