Prequantization Bundles Research Articles

Embedded contact homology of prequantization bundles

Embedded contact homology of prequantization bundles

Symplectic embeddings into disk cotangent bundles

In this paper, we compute the embedded contact homology (ECH) capacities of the disk cotangent bundles $D^*S^2$ and $D^*\mathbb{R} P^2$. We also find sharp symplectic embeddings into these domains. In particular, we compute their Gromov widths. In order to do that, we explicitly calculate the ECH chain complexes of $S^*S^2$ and $S^* \mathbb{R} P^2$ using a direct limit argument on the action inspired by Bourgeois's Morse-Bott approach and ideas from Nelson-Weiler's work on the ECH of prequantization bundles. Moreover, we use integrable systems techniques to find explicit symplectic embeddings. In particular, we prove that the disk cotangent bundles of a hemisphere and of a punctured sphere are symplectomorphic to an open ball and a symplectic bidisk, respectively.

Translated points for contactomorphisms of prequantization spaces over monotone symplectic toric manifolds

We prove a version of Sandon's conjecture on the number of translated points of contactomorphisms for the case of prequantization bundles over certain closed monotone symplectic toric manifolds. Namely we show that any contactomorphism of such a prequantization bundle lying in the identity component of the contactomorphism group possesses at least $N$ translated points, where $N$ is the minimal Chern number of the symplectic toric manifold. The proof relies on the theory of generating functions coupled with equivariant cohomology, whereby we adapt Givental's approach to the Arnold conjecture for integral symplectic toric manifolds to the context of prequantization bundles.

Symplectic structure for general relativity and Einstein–Brillouin–Keller quantization

The Hamiltonian system of general relativity and its quantization without any matter or gauge fields are discussed on the basis of the symplectic geometrical theory. A symplectic geometry of classical general relativity is constructed using a generalized phase space for pure gravity. Prequantization of the symplectic manifold is performed according to the standard procedure of geometrical quantization. Quantum vacuum solutions are chosen from among the classical solutions under the Einstein–Brillouin–Keller quantization condition. A topological correction of quantum solutions, namely the Maslov index, is realized using a prequantization bundle. In addition, a possible mass spectrum of Schwarzschild black holes is discussed.

Automatic transversality in contact homology II: filtrations and computations

This paper is the sequel to the previous paper [Nelson, Abh. Math. Semin. Univ. Hambg. 85 (2015) 125–179], which showed that sufficient regularity exists to define cylindrical contact homology in dimension three for nondegenerate dynamically separated contact forms, a subclass of dynamically convex contact forms. The Reeb orbits of these so-called dynamically separated contact forms satisfy a uniform growth condition on their Conley–Zehnder indices with respect to a free homotopy class. Given a contact form which is dynamically separated up to large action, we demonstrate a filtration by action on the chain complex and show how to obtain the desired cylindrical contact homology by taking direct limits. We give a direct proof of invariance of cylindrical contact homology within the class of dynamically separated contact forms, and elucidate the independence of the filtered cylindrical contact homology with respect to the choice of the dynamically separated contact form and almost complex structure. We also show that these regularity results are compatible with geometric methods of computing cylindrical contact homology of prequantization bundles, proving a conjecture of Eliashberg [Symplectic field theory and its applications, International Congress of Mathematicians I (European Mathematical Society, Zürich, 2007) 217–246] in dimension three.

Isotropic submanifolds and coadjoint orbits of the Hamiltonian group

We describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms of a symplectic manifold $(M,\omega)$ by implementing symplectic reduction for the dual pair associated to the Hamiltonian description of ideal fluids. The description is given in terms of nonlinear Grassmannians (manifolds of submanifolds) with additional geometric structures. Reduction at zero momentum yields the identification of coadjoint orbits with Grassmannians of isotropic volume submanifolds, slightly generalizing the results in Weinstein [1990] and Lee [2009]. At the other extreme, the case of a nondegenerate momentum recovers the identification of connected components of the nonlinear symplectic Grassmannian with coadjoint orbits, thereby recovering the result of Haller and Vizman [2004]. We also comment on the intermediate cases which correspond to new classes of coadjoint orbits. The description of these coadjoint orbits as well as their orbit symplectic form is obtained in a systematic way by exploiting the general properties of dual pairs of momentum maps. We also show that whenever the symplectic manifold $(M,\omega)$ is prequantizable, the coadjoint orbits that consist of isotropic submanifolds with total volume $a\in\mathbb{Z}$ are prequantizable. The prequantum bundle is constructed explicitly and, in the Lagrangian case, recovers the Berry bundle constructed in Weinstein [1990].

On the filtered symplectic homology of prequantization bundles

We study Reeb dynamics on prequantization circle bundles and the filtered (equivariant) symplectic homology of prequantization line bundles, aka negative line bundles, with symplectically aspherical base. We define (equivariant) symplectic capacities, obtain an upper bound on their growth, prove uniform instability of the filtered symplectic homology and touch upon the question of stable displacement. We also introduce a new algebraic structure on the positive (equivariant) symplectic homology capturing the free homotopy class of a closed Reeb orbit — the linking number filtration — and use it to give a new proof of the non-degenerate case of the contact Conley conjecture (i.e. the existence of infinitely many simple closed Reeb orbits), not relying on contact homology.

Multiplicity of closed Reeb orbits on prequantization bundles

We establish multiplicity results for geometrically distinct contractible closed Reeb orbits of non-degenerate contact forms on a broad class of prequantization bundles. The results hold under certain index requirements on the contact form and are sharp for unit cotangent bundles of CROSS’s. In particular, we generalize and put in the symplectic-topological context a theorem of Duan, Liu, Long and Wang for the standard contact sphere. We also prove similar results for non-hyperbolic contractible closed orbits and briefly touch upon the multiplicity problem for degenerate forms. On the combinatorial side of the question, we revisit and reprove the enhanced common jump theorem of Duan, Long and Wang, and interpret it as an index recurrence result.

Equivariant prequantization bundles on the space of connections and characteristic classes

We show how characteristic classes determine equivariant prequantization bundles over the space of connections on a principal bundle. These bundles are shown to generalize the Chern-Simons line bundles to arbitrary dimensions. Our result applies to arbitrary bundles, and it is studied the action of both the gauge group and the automorphisms group. The action of the elements in the connected component of the identity of the group generalizes known results in the literature. The action of the elements not connected with the identity is shown to be determined by a characteristic class by using differential characters and equivariant cohomology. We extend our results to the space of Riemannian metrics and the actions of diffeomorphisms. In dimension 2, a ${\Gamma}_{M}$-equivariant prequantization bundle of the Weil-Petersson symplectic form on the Teichm\"uller space is obtained, where ${\Gamma}_{M}$ is the mapping class group of the surface M.

The Conley–Zehnder indices of the Reeb flow action along $$S^1$$ S 1 -fibers over certain orbifolds

We prove a useful relation between the Conley–Zehnder indices of the Reeb vector flow action along periodic orbits in prequantization bundles and the orbifold Chern class of the base symplectic orbifolds motivated by the well-known case of manifolds. We also apply this method to primary examples.

Periodic Reeb orbits on prequantization bundles

In this paper, we prove that every graphical hypersurface in a prequantization bundle over a symplectic manifold $M$, pinched between two circle bundles whose ratio of radii is less than $\sqrt{2}$ carries either one short simple periodic orbit or carries at least $\operatorname{cuplength}(M)+1$ simple periodic Reeb orbits.

Dirac operators on quasi-Hamiltonian G-spaces

We develop notions of twisted spinor bundle and twisted pre-quantum bundle on quasi-Hamiltonian G-spaces. The main result of this paper is that we construct a Dirac operator with index given by positive energy representation of loop group. This generalizes the quantization of Hamiltonian $G$-spaces to quasi-Hamiltonian G-spaces.

On the Conley conjecture for Reeb flows

In this paper, we prove the existence of infinitely many closed Reeb orbits for a certain class of contact manifolds. This result can be viewed as a contact analogue of the Hamiltonian Conley conjecture. The manifolds for which the contact Conley conjecture is established are the pre-quantization circle bundles with aspherical base. As an application, we prove that for a surface of genus at least two with a non-vanishing magnetic field, the twisted geodesic flow has infinitely many periodic orbits on every low energy level.

$L_{\infty}$-algebras of local observables from higher prequantum bundles

To any manifold equipped with a higher degree closed form, one can associate an L-infinity algebra of local observables that generalizes the Poisson algebra of a symplectic manifold. Here, by means of an explicit homotopy equivalence, we interpret this L-infinity algebra in terms of infinitesimal autoequivalences of higher prequantum bundles. By truncating the connection data on the prequantum bundle, we produce analogues of the (higher) Lie algebras of sections of the Atiyah Lie algebroid and of the Courant Lie 2-algebroid. We also exhibit the L-infinity cocycle that realizes the L-infinity algebra of local observables as a Kirillov-Kostant-Souriau-type L-infinity extension of the Hamiltonian vector fields. When restricted along a Lie algebra action, this yields Heisenberg-like L-infinity algebras such as the string Lie 2-algebra of a semisimple Lie algebra.

EQUIVARIANT SYMPLECTIC HOMOLOGY AND MULTIPLE CLOSED REEB ORBITS

We study the existence of multiple closed Reeb orbits on some contact manifolds by means of S1-equivariant symplectic homology and the index iteration formula. We prove that a certain class of contact manifolds which admits displaceable exact contact embeddings, a certain class of prequantization bundles, and Brieskorn spheres have multiple closed Reeb orbits.

Toric Kähler metrics seen from infinity, quantization and compact tropical amoebas

We consider the metric space of all toric Kähler metrics on a compact toric manifold; when “looking at it from infinity” (following Gromov), we obtain the tangent cone at infinity, which is parametrized by equivalence classes of complete geodesics. In the present paper, we study the associated limit for the family of metrics on the toric variety, its quantization, and degeneration of generic divisors. The limits of the corresponding Kähler polarizations become degenerate along the Lagrangian fibration defined by the moment map. This allows us to interpolate continuously between geometric quantizations in the holomorphic and real polarizations and show that the monomial holomorphic sections of the prequantum bundle converge to Dirac delta distributions supported on Bohr-Sommerfeld fibers. In the second part, we use these families of toric metric degenerations to study the limit of compact hypersurface amoebas and show that in Legendre transformed variables they are described by tropical amoebas. We believe that our approach gives a different, complementary, perspective on the relation between complex algebraic geometry and tropical geometry.

Quillen bundle and geometric prequantization of non-abelian vortices on a Riemann surface

In this paper we prequantize the moduli space of non-abelian vortices. We explicitly calculate the symplectic form arising from L2 metric and we construct a prequantum line bundle whose curvature is proportional to this symplectic form. The prequantum line bundle turns out to be Quillen’s determinant line bundle with a modified Quillen metric. Next, as in the case of abelian vortices, we construct line bundles over the moduli space whose curvatures form a family of symplectic forms which are parametrized by Ψ0, a section of a certain bundle. The equivalence of these prequantum bundles are discussed.

A Lefschetz fixed point formula for symplectomorphisms

Consider a compact Kähler manifold endowed with a prequantum bundle. Following the geometric quantization scheme, the associated quantum spaces are the spaces of holomorphic sections of the tensor powers of the prequantum bundle. In this paper we construct an asymptotic representation of the prequantum bundle automorphism group in these quantum spaces. We estimate the characters of these representations under some transversality assumption. The formula obtained generalizes in some sense the Lefschetz fixed point formula for the automorphisms of the prequantum bundle preserving its holomorphic structure. Our results will be applied in two forthcoming papers to the quantum representation of the mapping class group.

On the geometric quantization of twisted Poisson manifolds

We study the geometric quantization process for twisted Poisson manifolds. First, we introduce the notion of Lichnerowicz-twisted Poisson cohomology for twisted Poisson manifolds and we use it in order to characterize their prequantization bundles and to establish their prequantization condition. Next, we introduce a polarization and we discuss the quantization problem. In each step, several examples are presented.

Semi-Classical Properties of Geometric Quantization with Metaplectic Correction

The geometric quantization of a symplectic manifold endowed with a prequantum bundle and a metaplectic structure is defined by means of an integrable complex structure. We prove that its semi-classical limit does not depend on the choice of the complex structure. We show this in two ways. First, by introducing unitary identifications between the quantum spaces associated to the various complex polarizations and second, by defining an asymptotically flat connection in the bundle of quantum spaces over the space of complex structures. Furthermore Berezin-Toeplitz operators are intertwined by these identifications and have principal and subprincipal symbols defined independently of the complex structure. The relation with the Schrödinger equation and the group of prequantum bundle automorphisms is considered as well.