A prequantization bundle is a circle bundle over a symplectic surface with negative Euler class. A connection 1-form induces a natural contact form on such a bundle. The purpose of this paper is to study symplectic embeddings of balls into the associated disk bundles. To this end, we compute the ECH cobordism maps induced by these disk bundles. In addition, we compute the ECH spectrum of the prequantization bundles over the sphere and the torus.
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 f…
This paper determines a condition that is necessary and sufficient for a metaplectic-c prequantizable symplectic manifold with an effective Hamiltonian torus action to admit an equivariant metaplectic-c prequantization. The condition is evaluated at a fixed point of the momentum map, and is shifted from the one that is known for equivariant prequantization line bundles. Given a metaplectic-c prequantized symplectic manifold with a Hamiltonian energy function, the author previously proposed a condition under which a regular value of the function should be considered a quantized energy level of …
This is a report for my Master's reading project where I review some basic ideas in the theory of prequantizing a symplectic manifold. The classic proof that a symplectic manifold is prequantizable if and only if its symplectic form is integral is given.
We define spin-c prequantization of a symplectic manifold to be a spin-c structure and a connection which are compatible with the symplectic form. We describe the cutting of an S^1-equivariant spin-c prequantization. The cutting process involves a choice of a spin-c prequantization for the complex plane. We prove that the cutting is possible if and only if the moment map level set along which the cutting is done is compatible with this choice.
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 …
We review the prequantization procedure in the context of super symplectic manifolds with a symplectic form which is not necessarily homogeneous. In developing the theory of non homogeneous symplectic forms, there is one surprising result: the Poisson algebra no longer is the set of smooth functions on the manifold, but a subset of functions with values in a super vector space of dimension 1|1. We show that this has no notable consequences for results concerning coadjoint orbits, momentum maps, and central extensions. Another surprising result is that prequantization in terms of complex line b…
In this paper we find connection between the Hofer's metric of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold, with an integral symplectic form, and the geometry, defined in a paper by Eliashberg and Polterovich, of the quantomorphisms group of its prequantization manifold. This gives two main results: First, we calculate, partly, the geometry of the quantomorphisms groups of a prequantization manifolds of an integral symplectic manifold which admits certain Lagrangian foliation. Second, for every prequantization manifold we give a formula for the distance between a p…
We generalize geometric prequantization of symplectic manifolds to differentiable stacks. Our approach is atlas-independent and provides a bijection between isomorphism classes of principal circle bundles (with or without connections) and second cohomology groups of certain chain complexes.
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.
The moduli space of solutions to the vortex equations on a Riemann surface are well known to have a symplectic (in fact Kähler) structure. We show this symplectic structure explictly and proceed to show a family of symplectic (in fact, Kähler) structures $Ω_{Ψ_0}$ on the moduli space, parametrised by $Ψ_0$, a section of a line bundle on the Riemann surface. Next we show that corresponding to these there is a family of prequantum line bundles ${\mathcal P}_{Ψ_0} $on the moduli space whose curvature is proportional to the symplectic forms $Ω_{Ψ_0}$.
In the classical Kostant-Souriau prequantization procedure, the Poisson algebra of a symplectic manifold $(M,ω)$ is realized as the space of infinitesimal quantomorphisms of the prequantization circle bundle. Robinson and Rawnsley developed an alternative to the Kostant-Souriau quantization process in which the prequantization circle bundle and metaplectic structure for $(M,ω)$ are replaced by a metaplectic-c prequantization. They proved that metaplectic-c quantization can be applied to a larger class of manifolds than the classical recipe. This paper presents a definition for a metaplectic-c …
A comparison on some facts concerning the geometric quantization of symplectic manifolds is presented here. Criticism, facts and improvements on the sophisticated theory of geometric quantization are presented touching briefly, all the "salient points of the theory". The unfamiliar reader can consider this as a "soft" introduction to the topic.
Classical mechanics has a natural mathematical setting in symplectic geometry and it may be asked if the same is true for quantum mechanics. More precisely, is it possible to capture certain quantum idiosyncrasies within the symplectic framework of classical mechanics? Beginning with Dirac, the idea of quantizing a classical system involved associating the phase space variables with Hermitian operators which act on some Hilbert space, as well as associating the Poisson bracket on the phase space with the commutator for the operators. Mathematically the phase space is associated with some sympl…
A 2-plectic manifold is a manifold equipped with a closed nondegenerate 3-form, just as a symplectic manifold is equipped with a closed nondegenerate 2-form. In 2-plectic geometry we meet higher analogues of many structures familiar from symplectic geometry. For example, any 2-plectic manifold has a Lie 2-algebra consisting of smooth functions and Hamiltonian 1-forms. This is equipped with a Poisson-like bracket which only satisfies the Jacobi identity up to `coherent chain homotopy'. Over any 2-plectic manifold is a vector bundle equipped with extra structure called an exact Courant algebroid…
Hitchin has shown that the moduli space ${\mathcal M}$ of the dimensionally reduced self-dual Yang-Mills equations has a hyperKähler structure. In this paper we first explicitly show the hyperKähler structure, the details of which is missing in Hitchin's paper. We show here that ${\mathcal M}$ admits three pre-quantum line bundles, corresponding to the three symplectic forms. We use Quillen's determinant line bundle construction and show that the Quillen curvatures of these prequantum line bundles are proportional to each of the symplectic forms mentioned above. The prequantum line bundles are…
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 combina…
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.…
The rational homology balls $B_n$ appeared in Fintushel and Stern's rational blow-down construction [FS2]. Later, Symington [Sy1], defined this operation in the symplectic category. In [Kh2], the author defined the inverse procedure, the symplectic rational blow-up. In this paper, we study the obstructions to symplectically rationally blowing up a symplectic 4-manifold, i.e. the obstructions to symplectically embedding the rational homology balls $B_n$ into a symplectic 4-manifold. We prove a theorem and give additional examples which suggest that in order to symplectically embed the rational …
In this paper we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian, S^2 or RP^2, in symplectic manifolds that are rational or ruled. Via a symplectic cutting construction this is a natural extension of McDuff's connectedness of ball packings in other settings and this result has applications to several different questions: smooth knotting and unknottedness results for spherical Lagrangians, the transitivity of the action of the symplectic Torelli group, classifying Lagrangian isotopy classes in the presence of knotting, and detecting Floer-theoreti…
In this paper we describe the intersection between the balls of maximal symplectic packings of $¶^2$. This analysis shows the existence of singular points for maximal packings of $¶^2$ by more than three equal balls. It also yields a construction of a class of very regular examples of maximal packings by five balls.
In this paper we obtain new obstructions to symplectic embeddings of the four-dimensional polydisk $P(a,1)$ into the ball $B(c)$ for $2\leq a<\frac{\sqrt{7}-1} {\sqrt{7}-2} \approx 2.549$, extending work done by Hind-Lisi and Hutchings. Schlenk's folding construction permits us to conclude our bound on $c$ is optimal. Our proof makes use of the combinatorial criterion necessary for one "convex toric domain" to symplectically embed into another introduced by Hutchings in \cite{Beyond}. Additionally, we prove that if certain symplectic embeddings of four dimensional convex toric domains exist th…
We compute in this note the full homotopy type of the space of symplectic embeddings of the standard ball $B^4(c) \subset \R^4$ (where $c= πr^2$ is the capacity of the standard ball of radius $r$) into the 4-dimensional rational symplectic manifold $M_μ= (S^2 \times S^2, μ\om_0 \oplus \om_0)$ where $\om_0$ is the area form on the sphere with total area 1 and $μ$ belongs to the interval $(1,2]$. We know, by the work of Lalonde-Pinsonnault, that this space retracts to the space of symplectic frames of $M_μ$ for any value of $c$ smaller than the critical value $μ-1$, and that its homotopy type do…
We study symplectic embeddings of ellipsoids into balls. In the main construction, we show that a given embedding of 2m-dimensional ellipsoids can be suspended to embeddings of ellipsoids in any higher dimension. In dimension 6,s if the ratio of the areas of any two axes is sufficiently large then the ellipsoid is flexible in the sense that it fully fills a ball. We also show that the same property holds in all dimensions for sufficiently thin ellipsoids E(1,..., a). A consequence of our study is that in arbitrary dimension a ball can be fully filled by any sufficiently large number of identic…
We survey some recent progress on understanding when one four-dimensional symplectic manifold can be symplectically embedded into another. In 2010, McDuff established a number-theoretic criterion for the existence of a symplectic embedding of one four-dimensional ellipsoid into another. This is related to previously known criteria for when a disjoint union of balls can be symplectically embedded into a ball. The new theory of "ECH capacities" gives general obstructions to symplectic embeddings in four dimensions which turn out to be sharp in the above cases.
We exhibit monotone Lagrangian tori inside the standard symplectic four-dimensional unit ball that become Hamiltonian isotopic to the Clifford torus, i.e.~the standard product torus, only when considered inside a strictly larger ball (they are not even symplectomorphic to a standard torus inside the unit ball). These tori are then used to construct new examples of symplectic embeddings of toric domains into the unit ball which are symplectically knotted in the sense of J.~Gutt and M.~Usher. We also give a characterisation of the Clifford torus inside the ball as well as the projective plane in…
Let M:=(M^{4},\om) be a 4-dimensional rational ruled symplectic manifold and denote by w_{M} its Gromov width. Let Emb_ω(B^{4}(c),M) be the space of symplectic embeddings of the standard ball B^4(c) \subset \R^4 of radius r and of capacity c:= πr^2 into (M,\om). By the work of Lalonde and Pinsonnault, we know that there exists a critical capacity \ccrit \in (0,w_{M}] such that, for all c\in(0,\ccrit), the embedding space Emb_ω(B^{4}(c),M) is homotopy equivalent to the space of symplectic frames \SFr(M). We also know that the homotopy type of Emb_ω(B^{4}(c),M) changes when c reaches \ccrit and …
In this note, we obtain new obstructions to symplectic embeddings of a product of disks (a polydisk) into a 4-dimensional ball. The polydisk P(r,s) is the product of the disk of area r with the disk of area s. The ball of capacity a, denoted B(a), is the ball with πr^2 \le a. We show P(1,2) embeds in B^4(a) if and only if a is at least 3. This shows the inclusion of P(1,2) in B^4(3) is optimal. The necessity of a \ge 3 implies that for this particular embedding problem neither the Ekeland-Hofer nor ECH capacities give a sharp obstruction. We contrast this with the case of ellipsoid embeddings …
We show how to reduce the problem of symplectically embedding one 4-dimensional rational ellipsoid into another to a problem of embedding disjoint unions of balls into appropriate blow ups of \C P^2. For example, the problem of embedding the ellipsoid E(1,k) into a ball B is equivalent to that of embedding k disjoint equal balls into \C P^2, and so can be solved by the work of Gromov, McDuff--Polterovich and Biran. (Here k is the ratio of the area of the major axis to that of the minor axis.) As a consequence we show that the ball may be fully filled by the ellipsoid E(1,k) for k=1,4 and all k…
We define and solve the toric version of the symplectic ball packing problem, in the sense of listing all 2n-dimensional symplectic-toric manifolds which admit a perfect packing by balls embedded in a symplectic and torus equivariant fashion. In order to do this we first describe a problem in geometric-combinatorics which is equivalent to the toric symplectic ball packing problem. Then we solve this problem using arguments from Convex Geometry and Delzant theory. Applications to symplectic blowing-up are also presented, and some further questions are raised in the last section.
We construct symplectic embeddings of ellipsoids of dimension $2n \ge 6$ into the product of a 4-ball or 4-dimensional cube with Euclidean space. A sequence of these embeddings can be shown to be optimal.
The aim of this paper is to explain a link between symplectic isotopies of open objects such as balls and flexibility properties of symplectic hypersurfaces. We get connectedness results for spaces of symplectic ellipsoids or maximal packings of $P^2$.
We prove that the non-squeezing theorem of Gromov holds for symplectomorphisms on an infinite-dimensional symplectic Hilbert space, under the assumption that the image of the ball is convex. The proof is based on the construction by duality methods of a symplectic capacity for bounded convex neighbourhoods of the origin. We also discuss some examples of symplectomorphisms on infinite-dimensional spaces exhibiting behaviours which would be impossible in finite dimensions.
Let M be a symplectic-toric manifold of dimension at least four. This paper investigates the so called symplectic ball packing problem in the toral equivariant setting. We show that the set of toric symplectic ball packings of M admits the structure of a convex polytope. Previous work of the first author shows that up to equivalence, only CP^1 x CP^1 and CP^2 admit density one packings when n=2 and only CP^n admits density one packings when n>2. In contrast, we show that for a fixed n>=2 and each r in (0, 1), there are uncountably many inequivalent 2n-dimensional symplectic-toric manifolds wit…
This note constructs sharp obstructions for stabilized symplectic embeddings of an ellipsoid into a ball, in the case when the initial four-dimensional ellipsoid has `eccentricity' of the form 3n-1 for some integer n.
We prove that the group of compactly supported symplectomorphisms of the standard symplectic ball admits a continuum of linearly independent real-valued homogeneous quasimorphisms. In addition these quasimorphisms are Lipschitz in the Hofer metric and have the following property: the value of each such quasimorphism on any symplectomorphism supported in any "sufficiently small" open subset of the ball equals the Calabi invariant of the symplectomorphism. By a "sufficiently small" open subset we mean that it can be displaced from itself by a symplectomorphism of the ball. As a byproduct we show…
McDuff and Schlenk determined when a four-dimensional ellipsoid can be symplectically embedded into a four-dimensional ball, and found that when the ellipsoid is close to round, the answer is given by an "infinite staircase" determined by the odd-index Fibonacci numbers. We show that this result still holds in higher dimensions when we "stabilize" the embedding problem.
We show that every 4-dimensional torus with a linear symplectic form can be fully filled by one symplectic ball. If such a torus is not symplectomorphic to a product of 2-dimensional tori with equal sized factors, then it can also be fully filled by any finite collection of balls provided only that their total volume is less than that of the 4-torus with its given linear symplectic form.
Define a "Liouville domain" to be a compact exact symplectic manifold with contact-type boundary. We use embedded contact homology to assign to each four-dimensional Liouville domain (or subset thereof) a sequence of real numbers, which we call "ECH capacities". The ECH capacities of a Liouville domain are defined in terms of the "ECH spectrum" of its boundary, which measures the amount of symplectic action needed to represent certain classes in embedded contact homology. Using cobordism maps on embedded contact homology (defined in joint work with Taubes), we show that the ECH capacities are …
We study the symplectic topology of some finite algebraic quotients of the An Milnor fibre which are diffeomorphic to the rational homology balls that appear in Fintushel and Stern's rational blowdown construction. We prove that these affine surfaces have no closed exact Lagrangian submanifolds by using the already available and deep understanding of the Fukaya category of the An Milnor fibre coming from homological mirror symmetry. On the other hand, we find Floer theoretically essential monotone Lagrangian tori, finitely covered by the monotone tori which we study in the An Milnor fibre. We …
Recently, McDuff and Schlenk determined the function c_{EB}(a) whose value at a is the infimum of the size of a 4-ball into which the ellipsoid E(1,a) symplectically embeds (here, a >= 1 is the ratio of the area of the large axis to that of the smaller axis of the ellipsoid). In this paper we look at embeddings into four-dimensional cubes instead, and determine the function c_{EC}(a) whose value at a is the infimum of the size of a 4-cube C^{4}(A) = D^{2}(A) times D^{2}(A) into which the ellipsoid E(1,a) symplectically embeds (where D^{2}(A) denotes the disc in mathbb{R}^{2} of area A). As in …
In any dimension $2n \ge 6$ we show that certain spaces of symplectic embeddings of a polydisk into a product $B^4 \times \Bbb R^{2(n-2)}$ of a $4$-ball and Euclidean space, are not path connected. We also show that any pair of such nonisotopic embeddings can never be extended to the same ellipsoid.
We apply the method of filling with holomorphic discs to a 4-dimensional symplectic cobordism with the standard contact 3-sphere as a convex boundary component. We establish the following dichotomy: either the cobordism is diffeomorphic to a ball, or there is a periodic Reeb orbit of quantifiably short period in the concave boundary of the cobordism. This allows us to give a unified treatment of various results concerning Reeb dynamics on contact 3-manifolds, symplectic fillability, the topology of symplectic cobordisms, symplectic non-squeezing, and the non-existence of exact Lagrangian surfa…
