--- title: Prequantization Bundles in Symplectic Geometry url: https://www.emergentmind.com/topics/prequantization-bundles type: topic --- # Prequantization Bundles in Symplectic Geometry A prequantization bundle is a principal circle bundle or, more generally, a geometric structure that encodes a necessary and sufficient condition for geometric quantization on a symplectic or related manifold. In its classical setup, if $(M,\omega)$ is a symplectic manifold with integral symplectic form, a prequantization bundle is a principal $S^1$–bundle $\pi:Y\to M$ endowed with a connection 1-form whose curvature pulls back to $\omega$. The existence and study of prequantization bundles is foundational in symplectic topology, contact geometry, low-dimensional topology, and mathematical physics, and has led to deep interactions with homological, Floer-theoretic, and categorical structures. ## 1. Definition and Basic Construction A prequantization bundle is constructed over a symplectic manifold $(M,\omega)$ with integral symplectic class $[\omega]\in H^2(M;\mathbb Z)$ as a principal $S^1$–bundle $\pi:Y\to M$ whose first Chern class satisfies $c_1(Y)=[\omega]$. It carries a connection 1-form $\alpha\in\Omega^1(Y)$ with $d\alpha = \pi^*\omega$. The Reeb vector field $X$ of $\alpha$ generates the $S^1$-action, and the triple $(Y,\alpha)$ is a contact manifold, often called a Boothby–Wang contact manifold [2007.13883, 2304.09652, 2406.08852]. The total space $Y$ can be equipped with a contact structure $\xi = \ker\alpha$; when $M$ is a closed symplectic manifold and $\omega$ is integral, $Y$ becomes a prequantum circle bundle. This construction also extends to negative line bundles (total space of a complex line bundle $E\rightarrow M$ with $c_1(E)=-[\omega]$), where the unit-circle subbundle with angular form yields the contact geometry [2304.09652, 1806.05705]. ## 2. Floer, ECH, and Rabinowitz Homology of Prequantization Bundles Prequantization bundles serve as central examples in the study of Floer homological invariants. For instance, in embedded contact homology (ECH), the chain complex for prequantization bundles over Riemann surfaces can be computed explicitly and relates to the base's homology algebra [2007.13883], while the ECH spectrum of such bundles can be described in terms of combinatorial data involving the Euler class and critical points of Morse functions on the base [2304.09652]. Rabinowitz Floer homology admits a split description for prequantization bundles, and there is a Floer–Gysin exact sequence relating Rabinowitz Floer homology of the prequantization boundary, the symplectic homology of the filling, and the quantum homology of the base [2311.17866]. These computations involve delicate Morse–Bott perturbations, direct-limit arguments using filtered Seiberg–Witten Floer cohomology, and a careful count of pseudoholomorphic curves, showing that the ECH (with certain gradings) is canonically isomorphic to the exterior algebra of the base surface’s homology [2007.13883]. The U-map action and its periodicity reflect deep features of the bundle’s contact topology. ## 3. Symplectic Fillings, Capacities, and Rigidity Strong symplectic fillings of prequantization bundles are subject to rigid classification if the base is aspherical and the filling satisfies finiteness of certain symplectic capacities. Specifically, for negative circle bundles over closed symplectic surfaces (with total space $Y$), a symplectically aspherical filling $(X,\Omega)$ whose capacity $c_1(X,\Omega)<\infty$ (in the sense of Siegel capacities/Gutt-Hutchings capacities) must be diffeomorphic to the standard disk bundle associated to the prequantization line bundle [2404.01105]. This result leverages holomorphic curve arguments and symplectic field theory techniques, showing the uniqueness of the filling type under mild topology and capacity constraints. Associated invariants such as filtered symplectic homology or S^1-equivariant symplectic homology (with additional algebraic structure from the linking number filtration) detect features like the stably displaceable nature of the zero-section and the existence of infinitely many closed Reeb orbits (Conley conjecture) in the prequantization total space [1806.05705]. ## 4. Reeb Dynamics and Contact Topology The Reeb dynamics on prequantization bundles are dictated entirely by the circle action: all Reeb orbits are fibers, and iterates thereof, with periods determined by the geometry. This setting provides a calibration for multiplicity problems in closed Reeb orbits: for a wide class of prequantization bundles, the minimal number of distinct contractible closed Reeb orbits is sharply governed by the Betti numbers and minimal Chern number of the base [1703.04054]. Explicit sharp lower bounds are established using symplectic-topological arguments and index recurrence results (enhanced common index jump theorem), showing that multiplicity results for Reeb orbits on unit cotangent bundles of symmetric spaces—viewed as prequantization bundles—are optimal. For non-hyperbolic orbits and for degenerate forms, further bounds are available, and the Conley–Zehnder indices of iterated orbits align in a manner that ensures each simple orbit's contribution is accounted for without overcounting [1703.04054]. ## 5. Prequantization in Moduli Spaces and Higher Structures Beyond classical symplectic manifolds, prequantization bundles extend to the moduli spaces of connections and solutions to gauge-theoretic equations (e.g., moduli of flat $\mathrm{PU}(p)$-bundles, or of the Seiberg–Witten equations). In these contexts, the prequantum line bundle is often realized as a Quillen-type determinant bundle or via Chern–Simons constructions, with curvature reproducing the geometric symplectic (or Kähler) form on the moduli space [2203.15997, 1012.4616, 1408.0253, 1703.05832]. The prequantization obstruction translates into the divisibility of cohomology classes or the vanishing of certain mappings in relative cohomology. In the context of groupoids and higher groupoids, the notion extends to Dixmier–Douady bundles (S^1-gerbes) classified by the integral 3-class, or even to bundle gerbes (and higher categorical analogues), culminating in the existence of prequantum 2-Hilbert spaces and their connection to higher geometric quantization and loop space transgression [1611.04711, 1608.08455, 2008.05184]. ## 6. Categorical and Homological Aspects Prequantization bundles underpin A∞-category and sheaf-theoretic perspectives in homological mirror symmetry and Floer theory. For integral symplectic manifolds, categories of rational (nonexact) Lagrangians have their Floer theory recoverable as exact Lagrangian Floer theory in fillings of the associated prequantization bundle, enabling Fukaya–sheaf correspondences and categorical liftings of quantum cohomology [2406.08852]. The structure of prequantum line bundle gerbes allows for 2-Hilbert spaces of sections, with monoidal, semisimple, and abelian properties, matching categorical expectations for higher prequantization [1608.08455]. These categorical structures preserve and generalize Kostant–Souriau prequantization, generating representations not just on Hilbert spaces, but on categories or 2-categories of sections, compatible under morphisms, dualities, and monoidal structures. ## 7. Quantization via Deformation and Extensions Prequantization bundles provide a geometric setting for deformation quantization: the "Souriau bracket" on the prequantum bundle enables a deformation of $C^\infty(Y)$ giving rise to a quantum product that reflects both the bundle geometry and the underlying Poisson structure [1108.1385]. This structure ensures that the module of polarized wave-functions inherits an operator representation compatible with the algebraic star-product, tying together geometric and deformation quantization in a single framework. Extensions to 2-plectic geometry and higher gerbes yield "prequantization" for closed 3-forms, utilizing $PU(H)$-bundles and bundle gerbes equipped with connections and curving, with explicit Lie 2-algebra morphisms realizing the quantization map [2008.05184, 1608.08455]. --- ### References - [2007.13883]: Embedded contact homology of prequantization bundles - [2311.17866]: Rabinowitz Floer homology for prequantization bundles and Floer Gysin sequence - [2304.09652]: ECH spectrum of some prequantization bundles - [2404.01105]: On aspherical symplectic fillings with finite capacities of the prequantization bundles - [1806.05705]: On the Filtered Symplectic Homology of Prequantization Bundles - [2406.08852]: On Fukaya categories and prequantization bundles - [1703.04054]: Multiplicity of Closed Reeb Orbits on Prequantization Bundles - [2203.15997]: Quillen-type bundle and geometric prequantization on moduli space of the Seiberg-Witten equations on product of Riemann surfaces - [1012.4616]: Quillen bundle and Geometric Prequantization of Non-Abelian Vortices on a Riemann surface - [1408.0253]: Prequantization of the Moduli Space of Flat ${\rm PU}(p)$-Bundles with Prescribed Boundary Holonomies - [1611.04711]: Groupoid equivariant prequantization - [1608.08455]: The 2-Hilbert Space of a Prequantum Bundle Gerbe - [2008.05184]: On the prequantisation map for 2-plectic manifolds - [1108.1385]: Quantization via Deformation of Prequantization - [1703.05832]: Equivariant prequantization bundles on the space of connections and characteristic classes Source: https://www.emergentmind.com/topics/prequantization-bundles