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Showing papers on "Symplectic vector space published in 2014"


Journal ArticleDOI
TL;DR: In this paper, the authors established new restrictions on the embedding of basic shapes in symplectic vector spaces and showed that these restrictions are sharp and can be improved by refining an embedding technique due to Guth.
Abstract: In this paper we establish new restrictions on the symplectic embeddings of basic shapes in symplectic vector spaces. By refining an embedding technique due to Guth, we also show that they are sharp.

60 citations


Journal ArticleDOI
TL;DR: In this article, a covariant algebra of observables is proposed for quantization of connections on arbitrary principal U(1)-bundles over globally hyperbolic Lorentzian manifolds.
Abstract: The aim of this work is to complete our program on the quantization of connections on arbitrary principal U(1)-bundles over globally hyperbolic Lorentzian manifolds. In particular, we show that one can assign via a covariant functor to any such bundle an algebra of observables which separates gauge equivalence classes of connections. The C*-algebra we construct generalizes the usual CCR-algebras, since, contrary to the standard field-theoretic models, it is based on a presymplectic Abelian group instead of a symplectic vector space. We prove a no-go theorem according to which neither this functor, nor any of its quotients, satisfies the strict axioms of general local covariance. As a byproduct, we prove that a morphism violates the locality axiom if and only if a certain induced morphism of cohomology groups is non-injective. We show then that, fixing any principal U(1)-bundle, there exists a suitable category of subbundles for which a quotient of our functor yields a quantum field theory in the sense of Haag and Kastler. We shall provide a physical interpretation of this feature and we obtain some new insights concerning electric charges in locally covariant quantum field theory.

49 citations


Journal ArticleDOI
TL;DR: A symplectic integrator, based on the implicit midpoint method, for classical spin systems where each spin is a unit vector in R{3}, yields a new integrable discretization of the spinning top.
Abstract: We present a symplectic integrator, based on the implicit midpoint method, for classical spin systems where each spin is a unit vector in R-3. Unlike splittingmethods, it is defined for all Hamiltonians and is O(3)-equivariant, i.e., coordinate-independent. It is a rare example of a generating function for symplectic maps of a noncanonical phase space. It yields a new integrable discretization of the spinning top.

33 citations


Journal ArticleDOI
TL;DR: In this paper, derived algebraic geometry is used to construct topological field theories with values in higher categories of Lagrangian correspondences, including moment maps, quasi-Hamiltonian structures, and mapping stacks with boundary conditions.

30 citations


Journal ArticleDOI
TL;DR: The generalized Conley-Zehnder index (GZ index) as discussed by the authors is based on the Maslov-type index and is defined for a continuous path of Lagrangians in a symplectic vector space.
Abstract: The Conley-Zehnder index associates an integer to any continuous path of symplectic matrices starting from the identity and ending at a matrix which does not admit 1 as an eigenvalue. We give new ways to compute this index. Robbin and Salamon define a generalization of the Conley-Zehnder index for any continuous path of symplectic matrices; this generalization is half integer valued. It is based on a Maslov-type index that they define for a continuous path of Lagrangians in a symplectic vector space $(W,\bar{\Omega})$, having chosen a given reference Lagrangian $V$. Paths of symplectic endomorphisms of $(\R^{2n},\Omega_0)$ are viewed as paths of Lagrangians defined by their graphs in $(W=\R^{2n}\oplus \R^{2n},\bar{\Omega}=\Omega_0\oplus -\Omega_0)$ and the reference Lagrangian is the diagonal. Robbin and Salamon give properties of this generalized Conley-Zehnder index and an explicit formula when the path has only regular crossings. We give here an axiomatic characterization of this generalized Conley-Zehnder index. We also give an explicit way to compute it for any continuous path of symplectic matrices.

30 citations


Journal ArticleDOI
TL;DR: In this article, the authors prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian in the presence of rational or ruled manifolds, via a symplectic cutting construction.
Abstract: In this paper, we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian, $$S^{2}$$ or $$\mathbb{RP }^{2}$$ , 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-theoretically essential Lagrangian tori in the del Pezzo surfaces.

29 citations


Journal ArticleDOI
TL;DR: In this article, the authors give finiteness results and some classifications up to diffeomorphism of minimal strong symplectic fillings of Seifert fibered spaces over S^2 satisfying certain conditions, with a fixed natural contact structure.
Abstract: We give finiteness results and some classifications up to diffeomorphism of minimal strong symplectic fillings of Seifert fibered spaces over S^2 satisfying certain conditions, with a fixed natural contact structure. In some cases we can prove that all symplectic fillings are obtained by rational blow-downs of a plumbing of spheres. In other cases, we produce new manifolds with convex symplectic boundary, thus yielding new cut-and-paste operations on symplectic manifolds containing certain configurations of symplectic spheres.

21 citations


Posted Content
TL;DR: In this article, the authors studied the action of symplectic homeomorphisms on smooth submanifolds, with a main focus on the behaviour of the homeomorphism with respect to numerical invariants like capacities.
Abstract: This paper studies the action of symplectic homeomorphisms on smooth submanifolds, with a main focus on the behaviour of symplectic homeomorphisms with respect to numerical invariants like capacities. Our main result is that a symplectic homeomorphism may preserve and squeeze codimension $4$ symplectic submanifolds ($C^0$-flexibility), while this is impossible for codimension $2$ symplectic submanifolds ($C^0$-rigidity). We also discuss $C^0$-invariants of coistropic and Lagrangian submanifolds, proving some rigidity results and formulating some conjectures. We finally formulate an Eliashberg-Gromov $C^0$-rigidity type question for submanifolds, which we solve in many cases. Our main technical tool is a quantitative $h$-principle result in symplectic geometry.

20 citations


Journal ArticleDOI
TL;DR: In this paper, the Schlafli identity is examined from a symplectic and semiclassical standpoint in the special case of flat, 3-dimensional space, and a proof is given, based on symplectic geometry.
Abstract: The Schlafli identity, which is important in Regge calculus and loop quantum gravity, is examined from a symplectic and semiclassical standpoint in the special case of flat, 3-dimensional space. In this case a proof is given, based on symplectic geometry. A series of symplectic and Lagrangian manifolds related to the Schlafli identity, including several versions of a Lagrangian manifold of tetrahedra, are discussed. Semiclassical interpretations of the various steps are provided. Possible generalizations to 3-dimensional spaces of constant (nonzero) curvature, involving Poisson-Lie groups and q-deformed spin networks, are discussed.

19 citations


Journal ArticleDOI
TL;DR: In this article, a new two-level finite element method using the symplectic series as global functions while using the conventional finite element shape functions as local functions is developed, and a number of numerical examples as well as convergence studies are given.

19 citations


Journal ArticleDOI
TL;DR: In this paper, a construction of 81 symplectic resolutions of a 4-dimensional quotient singularity obtained by an action of a group of order 32 is presented. But the existence of such resolutions is known by a result of Bellamy and Schedler.
Abstract: We provide a construction of 81 symplectic resolutions of a 4-dimensional quotient singularity obtained by an action of a group of order 32. The existence of such resolutions is known by a result of Bellamy and Schedler. Our explicit construction is obtained via GIT quotient of the spectrum of a ring graded in the Picard group generated by the divisors associated to the conjugacy classes of symplectic reflections of the group in question. As the result we infer the geometric structure of these resolutions and their flops. Moreover, we represent the group in question as a group of automorphisms of an abelian 4-fold so that the resulting quotient has singularities with symplectic resolutions. This yields a new Kummer-type symplectic 4-fold.

Journal ArticleDOI
24 Nov 2014
TL;DR: In this article, a class of smooth torus manifolds whose orbit space has the struc- ture of a simple polytope with holes was studied and it was shown that these manifolds have stable almost complex structure and gave combinatorial formula for some of their Hirzebruch genera.
Abstract: We study a class of smooth torus manifolds whose orbit space has the struc- ture of a simple polytope with holes. We prove that these manifolds have stable almost complex structure and give combinatorial formula for some of their Hirzebruch genera. They have (invariant) almost complex structure if they admit positive omniorientation. In dimension four, we calculate the homology groups, construct symplectic structure on a large class of these manifolds, and give a family which is symplectic but not complex.

Journal ArticleDOI
TL;DR: In this article, a self contained and elementary description of normal forms for symplectic matrices, based on geometrical considerations, is given, which are expressed in terms of elementary Jordan matrices and integers with values in {-1, 0, 1} related to signatures of quadratic forms naturally associated to the symplectic matrix.
Abstract: We give a self contained and elementary description of normal forms for symplectic matrices, based on geometrical considerations. The normal forms in question are expressed in terms of elementary Jordan matrices and integers with values in {-1, 0, 1} related to signatures of quadratic forms naturally associated to the symplectic matrix. © European Mathematical Society.

Journal ArticleDOI
TL;DR: In this paper, the authors compared the universal Poisson deformation of the symplectic quotient singularity with the deformation given by the Calogero-Moser space.
Abstract: Let $\Gamma$ be a finite subgroup of $\mathrm{Sp}(V)$. In this article we count the number of symplectic resolutions admitted by the quotient singularity $V / \Gamma$. Our approach is to compare the universal Poisson deformation of the symplectic quotient singularity with the deformation given by the Calogero-Moser space. In this way, we give a simple formula for the number of $\mathbb{Q}$-factorial terminalizations admitted by the symplectic quotient singularity in terms of the dimension of a certain Orlik-Solomon algebra naturally associated to the Calogero-Moser deformation. This dimension is explicitly calculated for all groups $\Gamma$ for which it is known that $V / \Gamma$ admits a symplectic resolution. As a consequence of our results, we confirm a conjecture of Ginzburg and Kaledin.

Journal ArticleDOI
TL;DR: In this paper, the authors study various aspects of the metaplectic Howe duality realized by the Fischer decomposition for the representation space of polynomials on R 2 n valued in the Segal-Shale-Weil representation.

Journal ArticleDOI
TL;DR: In this paper, a simply connected and non-simply connected Calabi-Yau 6-manifolds with fundamental groups Z p × Z q, and Z × Q q for any p ≥ 1 and q ≥ 2 via co-isotropic Luttinger surgery were constructed.

Journal ArticleDOI
TL;DR: In this article, the authors consider a curve of Fredholm pairs of Lagrangian subspaces in a fixed Banach space with continuously varying (weak) symplectic structures and derive a desuspension spectral flow formula for varying well-posed boundary conditions on manifolds with boundary.
Abstract: We consider a curve of Fredholm pairs of Lagrangian subspaces in a fixed Banach space with continuously varying (weak) symplectic structures. Assuming vanishing index, we obtain intrinsically a continuously varying splitting of the total Banach space into pairs of symplectic subspaces. Using such decompositions we define the curve's Maslov index by symplectic reduction to the classical finite-dimensional case. We prove the transitivity of repeated symplectic reductions and obtain the invariance of the Maslov index under symplectic reduction, while recovering all the standard properties of the Maslov index. As an application, we consider curves of elliptic operators which have varying principal symbol, varying maximal domain and are not necessarily of Dirac type. For this class of operator curves, we derive a desuspension spectral flow formula for varying well-posed boundary conditions on manifolds with boundary and obtain the splitting of the spectral flow on partitioned manifolds.

Journal ArticleDOI
TL;DR: In this paper, the authors investigated connections between the number of focal points of their recessive solutions of the Riccati matrix difference equations and the comparative index theory for discrete symplectic systems.
Abstract: We present new relations connecting the number of focal points of conjoined bases of eventually disconjugate symplectic difference systems with different coefficient matrices which obey the so-called majorant condition at For the case of controllable (near ) symplectic systems we investigate connections between the number of focal points of their recessive solutions. The results of the paper are based on the concept of minimal and maximal solutions of the associated Riccati matrix difference equations and the comparative index theory for discrete symplectic systems.

Journal ArticleDOI
TL;DR: In this article, the Nagata bound on the number of maximal Lagrangian subbundles of a general symplectic bundle has been shown to be equivalent to the Hirschowitz bound for vector bundles.
Abstract: A symplectic or orthogonal bundle V of rank 2n over a curve has an invariant t(V) which measures the maximal degree of its isotropic subbundles of rank n. This invariant t defines stratifications on moduli spaces of symplectic and orthogonal bundles. We study this stratification by relating it to another one given by secant varieties in certain extension spaces. We give a sharp upper bound on t(V), which generalizes the classical Nagata bound for ruled surfaces and the Hirschowitz bound for vector bundles, and study the structure of the stratifications on the moduli spaces. In particular, we compute the dimension of each stratum. We give a geometric interpretation of the number of maximal Lagrangian subbundles of a general symplectic bundle, when this is finite. We also observe some interesting features of orthogonal bundles which do not arise for symplectic bundles, essentially due to the richer topological structure of the moduli space in the orthogonal case.

Journal ArticleDOI
TL;DR: In this paper, the authors presented an integrator for Lie-Poisson systems with a sufficiently large symmetry group acting on the fibres of the manifold, and generalizes to the case that the vector space carries a bifoliation.
Abstract: We construct symplectic integrators for Lie–Poisson systems. The integrators are standard symplectic (partitioned) Runge–Kutta methods. Their phase space is a symplectic vector space equipped with a Hamiltonian action with momentum map J whose range is the target Lie–Poisson manifold, and their Hamiltonian is collective, that is, it is the target Hamiltonian pulled back by J. The method yields, for example, a symplectic midpoint rule expressed in 4 variables for arbitrary Hamiltonians on $\mathfrak{so}(3)^*$ . The method specializes in the case that a sufficiently large symmetry group acts on the fibres of J, and generalizes to the case that the vector space carries a bifoliation. Examples involving many classical groups are presented.

Posted Content
TL;DR: In this article, the authors investigated the notion of compactifying divisors for open algebraic surfaces and gave a sufficient and necessary criterion, which is simple and also works in higher dimensions, to determine whether an arbitrarily small concave/convex neighborhood exist for an $\omega$-orthogonal symplectic divisor (a symplectic plumbing).
Abstract: We investigate the notion of symplectic divisorial compactification for symplectic 4-manifolds with either convex or concave type boundary. This is motivated by the notion of compactifying divisors for open algebraic surfaces. We give a sufficient and necessary criterion, which is simple and also works in higher dimensions, to determine whether an arbitrarily small concave/convex neighborhood exist for an $\omega$-orthogonal symplectic divisor (a symplectic plumbing). If deformation of symplectic form is allowed, we show that a symplectic divisor has either a concave or convex neighborhood whenever the symplectic form is exact on the boundary of its plumbing. As an application, we classify symplectic compactifying divisors having finite boundary fundamental group. We also obtain a finiteness result of fillings when the boundary can be capped by a symplectic divisor with finite boundary fundamental group.

Journal ArticleDOI
TL;DR: In this paper, it was shown that two monotone Lagrangian submanifolds embedded in the standard symplectic vector space with the same monotonicity constant cannot link one another and that, individually, their smooth knot type is determined entirely by the homotopy theoretic data which classifies the underlying Lagrangians immersion.
Abstract: Under certain topological assumptions, we show that two monotone Lagrangian submanifolds embedded in the standard symplectic vector space with the same monotonicity constant cannot link one another and that, individually, their smooth knot type is determined entirely by the homotopy theoretic data which classifies the underlying Lagrangian immersion. The topological assumptions are satisfied by a large class of manifolds which are realised as monotone Lagrangians, including tori. After some additional homotopy theoretic calculations, we deduce that all monotone Lagrangian tori in the symplectic vector space of odd complex dimension at least five are smoothly isotopic.

Journal ArticleDOI
TL;DR: In this article, the Hilbert-Chow morphism was used to construct a canonical desingularization of the symplectic reduction, which is isomorphic to the closure of a nilpotent orbit in a simple Lie algebra.
Abstract: Let $$G \subset GL(V)$$ be a reductive algebraic subgroup acting on the symplectic vector space $$W=(V \oplus V^*)^{\oplus m}$$ , and let $$\mu :\ W \rightarrow Lie(G)^*$$ be the corresponding moment map. In this article, we use the theory of invariant Hilbert schemes to construct a canonical desingularization of the symplectic reduction $$\mu ^{-1}(0)/\!/G$$ for classes of examples where $$G=GL(V)$$ , $$O(V)$$ , or $$Sp(V)$$ . For these classes of examples, $$\mu ^{-1}(0)/\!/G$$ is isomorphic to the closure of a nilpotent orbit in a simple Lie algebra, and we compare the Hilbert–Chow morphism with the (well-known) symplectic desingularizations of $$\mu ^{-1}(0)/\!/G$$ .

Dissertation
27 Jun 2014
TL;DR: In this article, the minimal number of distinct Reeb orbits on a contact manifold which is the boundary of a compact manifold with contact type boundary is studied. But the authors focus on the invariance of the homologies with respect to the choice of the contact form on the boundary.
Abstract: This thesis deals with the question of the minimal number of distinct periodic Reeb orbits on a contact manifold which is the boundary of a compact symplectic manifold.The positive S1-equivariant symplectic homology is one of the main tools considered in this thesis. It is built from periodic orbits of Hamiltonian vector fields in a symplectic manifold whose boundary is the given contact manifold.Our first result describes the relation between the symplectic homologies of an exact compact symplectic manifold with contact type boundary (also called Liouville domain), and the periodic Reeb orbits on the boundary. We then prove some properties of these homologies. For a Liouville domain embedded into another one, we construct a morphism between their homologies. We study the invariance of the homologies with respect to the choice of the contact form on the boundary.We use the positive S1-equivariant symplectic homology to give a new proof of a Theorem by Ekeland and Lasry about the minimal number of distinct periodic Reeb orbits on some hypersurfaces in R2n. We indicate how it extends to some hypersurfaces in some negative line bundles. We also give a characterisation and a new way to compute the generalized Conley-Zehnder index defined by Robbin and Salamon for any path of symplectic matrices. A tool for this is a new analysis of normal forms for symplectic matrices.

Posted Content
TL;DR: In this article, the authors studied different aspects of the curvature flow on locally homogeneous manifolds, including long-time existence, solitons, regularity and convergence, and developed two classes of Lie groups, which are relatively simple from a structural point of view but geometrically rich and exotic: solvable Lie groups with a codimension one abelian normal subgroup and a construction attached to each left symmetric algebra.
Abstract: Recently, J. Streets and G. Tian introduced a natural way to evolve an almost-Kahler manifold called the symplectic curvature flow, in which the metric, the symplectic structure and the almost-complex structure are all evolving. We study in this paper different aspects of the flow on locally homogeneous manifolds, including long-time existence, solitons, regularity and convergence. We develop in detail two classes of Lie groups, which are relatively simple from a structural point of view but yet geometrically rich and exotic: solvable Lie groups with a codimension one abelian normal subgroup and a construction attached to each left symmetric algebra. As an application, we exhibit a soliton structure on most of symplectic surfaces which are Lie groups. A family of ancient solutions which develop a finite time singularity was found; neither their Chern scalar nor their scalar curvature are monotone along the flow and they converge in the pointed sense to a (non-Kahler) shrinking soliton solution on the same Lie group.

Journal ArticleDOI
TL;DR: The maximal weight of a conical symplectic variety X is, by definition, the maximal weight for the minimal homogeneous generators of the coordinate ring R of X, up to isomorphism as discussed by the authors.
Abstract: For positive integers N and d, there are only finite number of conical symplectic varieties of dimension 2d with maximal weights N, up to isomorphism. The maximal weight of a conical symplectic variety X is, by definition, the maximal weight of the minimal homogeneous generators of the coordinate ring R of X.

Journal ArticleDOI
TL;DR: In this article, it was shown that the kernels of the restrictions of Dirac-Dolbeault operators on natural subspaces of polynomial valued spinor fields are finite dimensional on a compact symplectic manifold.

Posted Content
TL;DR: In this article, various stability results for symplectic surfaces in symplectic $4-$manifolds with $b^+=1$ were established and applied to prove the existence of representatives of Lagrangian ADE-configurations as well as to classify negative symplectic spheres.
Abstract: We establish various stability results for symplectic surfaces in symplectic $4-$manifolds with $b^+=1$. These results are then applied to prove the existence of representatives of Lagrangian ADE-configurations as well as to classify negative symplectic spheres in symplectic $4-$manifolds with $\kappa=-\infty$. This involves the explicit construction of spheres in rational manifolds via a new construction technique called the tilted transport.

Journal ArticleDOI
TL;DR: The SR factorization for a given matrix A is a QR-like factorization A=SR, where the matrix S is symplectic and R is J-upper triangular, and the MSGS is the symplectic Gram-Schmidt algorithm implemented via the SGS.
Abstract: The SR factorization for a given matrix A is a QR-like factorization A=SR, where the matrix S is symplectic and R is J-upper triangular. This factorization is fundamental for some important structure-preserving methods in linear algebra and is usually implemented via the symplectic Gram-Schmidt algorithm (SGS).

Posted Content
TL;DR: In this article, the authors explore a hyperkahler analogue of Guillemin, Jeffrey and Sjamaar's construction of symplectic implosion (13), which may be viewed as an abelianisation procedure: given a symplectic man-ifold M with a Hamiltonian action of a compact group K, the implosion Mimpl is a new symplectic space with an action of the maximal torus T of K, such that the symplectic reductions of Mimpl by T agree with the reduction of M by K.
Abstract: Hyperkahler manifolds occupy a special position at the intersection of Riemannian, symplectic and algebraic geometry. A hyperkahler structure involves a Riemannian metric, as well as a triple of complex structures satisfying the quaternionic relations. Moreover we require that the metric is Kahler with respect to each complex structure, so we have a triple (in fact a whole two-sphere) of symplectic forms. Of course, there is no Darboux theorem in hyperkahler geometry because the metric contains local information. However, many of the construc- tions and results of symplectic geometry, especially those related to moment maps, do have analogues in the hyperkahler world. The pro- totype is the hyperkahler quotient construction (15), and more recent examples include hypertoric varieties (3) and cutting (9). In this article we shall explore a hyperkahler analogue of Guillemin, Jeffrey and Sjamaar's construction of symplectic implosion (13). This may be viewed as an abelianisation procedure: given a symplectic man- ifold M with a Hamiltonian action of a compact group K, the implosion Mimpl is a new symplectic space with an action of the maximal torus T of K, such that the symplectic reductions of Mimpl by T agree with the reductions of M by K. However the implosion is usually not smooth but is a singular space with a stratified symplectic structure. The im- plosion of the cotangent bundle T ∗ K acts as a universal object here; implosions of general Hamiltonian K-manifolds may be defined using the symplectic implosion (T ∗ K)impl. This space also has an algebro- geometric description as the geometric invariant theory quotient of KC by a maximal unipotent subgroup N. In (7) we introduced a hyperkahler analogue of the universal im- plosion in the case of SU(n) actions. The construction proceeds via quiver diagrams, and produces a stratified hyperkahler space Q. The hyperkahler strata can be described in terms of open sets in complex symplectic quotients of the cotangent bundle of KC = SL(n,C) by subgroups containing commutators of parabolic subgroups. There is a maximal torus action, and hyperkahler quotients by this action yield