Showing papers on "Symplectic vector space published in 2009"
TL;DR: In this paper, the strong form of Heisenberg inequalities due to Robertson and Schrodinger can be formally derived using only classical considerations, using a statistical tool known as the minimum volume ellipsoid together with the notion of symplectic capacity, which they view as a topological measure of uncertainty invariant under Hamiltonian dynamics.
Abstract: We show that the strong form of Heisenberg’s inequalities due to Robertson and Schrodinger can be formally derived using only classical considerations. This is achieved using a statistical tool known as the “minimum volume ellipsoid” together with the notion of symplectic capacity, which we view as a topological measure of uncertainty invariant under Hamiltonian dynamics. This invariant provides a right measurement tool to define what “quantum scale” is. We take the opportunity to discuss the principle of the symplectic camel, which is at the origin of the definition of symplectic capacities, and which provides an interesting link between classical and quantum physics.
90 citations
TL;DR: In this article, it was shown that for each k>3 there are infinitely many finite type Stein manifolds diffeomorphic to Euclidean space ℝ2k which are pairwise distinct as symplectic manifolds.
Abstract: We show that for each k>3 there are infinitely many finite type Stein manifolds diffeomorphic to Euclidean space ℝ2k which are pairwise distinct as symplectic manifolds.
90 citations
TL;DR: In this article, it was shown that the Yoneda coupling with the semiregularity map is a closed 2-form on moduli spaces of sheaves, which is a symplectic form.
88 citations
TL;DR: In this paper, the authors studied non-elliptic quadratic differential operators and proved that the spectrum of the operator is discrete and can be described as in the case of global ellipticity.
Abstract: We study non-elliptic quadratic differential operators. Quadratic differential operators are non-selfadjoint operators defined in the Weyl quantization by complex-valued quadratic symbols. When the real part of their Weyl symbols is a non-positive quadratic form, we point out the existence of a particular linear subspace in the phase space intrinsically associated to their Weyl symbols, called a singular space, such that when the singular space has a symplectic structure, the associated heat semigroup is smoothing in every direction of its symplectic orthogonal space. When the Weyl symbol of such an operator is elliptic on the singular space, this space is always symplectic and we prove that the spectrum of the operator is discrete and can be described as in the case of global ellipticity. We also describe the large time behavior of contraction semigroups generated by these operators.
82 citations
TL;DR: In this article, a method to construct singular Lagrangian 3-torus fibrations over certain a priori given integral affine manifolds with singularities is presented.
Abstract: We prove that Mark Gross' [8] topological Calabi-Yau compactifications can be made into symplectic compactifications. To prove this we develop a method to construct singular Lagrangian 3-torus fibrations over certain a priori given integral affine manifolds with singularities, which we call simple. This produces pairs of compact symplectic 6-manifolds homeomorphic to mirror pairs of Calabi-Yau 3-folds together with Lagrangian fibrations whose underlying integral affine structures are dual.
52 citations
TL;DR: In this paper, a curvature inequality for SO(3)-bundle with connection was studied for manifolds with dimension four, where the base has dimension four and the manifold is a manifold with dimension c 1 = 0.
Abstract: Given an SO(3)-bundle with connection, the associated two- sphere bundle carries a natural closed 2-form. Asking that this be symplectic gives a curvature inequality first considered by Rezn- ikov [34]. We study this inequality in the case when the base has dimension four, with three main aims. Firstly, we use this approach to construct symplectic six-manifolds with c 1 =0 which are never Kahler; e.g., we produce such manifolds with b 1 = 0 = b 3 and also with c 2 [w]
47 citations
01 Jan 2009
TL;DR: In this article, a general approach to unitary representations for all Lie groups is introduced, where the underlying feature is a study of sympletic manifolds X 2n (i.e., there exists a closed non-singular 2-form on X).
Abstract: We introduce a general approach to unitary representations for all Lie groups. An underlying feature is a study of sympletic manifolds X 2n (i. e. there exists a closed non-singular 2-form on X). If [?] ? H 2(X, R) is an integral class there is an associated affinely connected Hermitian line bundle L over X which is unique if X is simply connected.
41 citations
TL;DR: In this paper, the authors extend the results of that paper to construct, for certain Calabi-Yau A ∞ categories, something playing the role of the Gromov-Witten potential.
Abstract: This is the sequel to my paper ‘TCFTs and Calabi-Yau categories’, Advances in Mathematics 210 (2007) no. 1, 165-214. Here we extend the results of that paper to construct, for certain Calabi-Yau A ∞ categories, something playing the role of the Gromov-Witten potential. This is a state in the Fock space associated to periodic cyclic homology, which is a symplectic vector space. Applying this to a suitable A ∞ version of the derived category of sheaves on a Calabi-Yau yields the B model potential, at all genera. The construction does not go via the Deligne-Mumford spaces, but instead uses the Batalin-Vilkovisky algebra constructed from the uncompactified moduli spaces of curves by Sen and Zwiebach. The fundamental class of Deligne-Mumford space is replaced here by a certain solution to the quantum master equation, essentially the ‘string vertices’ of Zwiebach. On the field theory side, the BV operator has an interpretation as the quantized differential on the Fock space for periodic cyclic chains. Passing to homology, something satisfying the master equation yields an element of the Fock space.
36 citations
TL;DR: In this article, a method to obtain resolutions of symplectic orbifolds arising from symplectic reduction of a Hamiltonian S^1-manifold at a regular value is presented.
Abstract: In this paper we present a method to obtain resolutions of symplectic orbifolds arising from symplectic reduction of a Hamiltonian S^1-manifold at a regular value. As an application, we show that all isolated cyclic singularities of a symplectic orbifold admit a resolution and that pre-quantisations of symplectic orbifolds are symplectically fillable by a smooth manifold.
35 citations
TL;DR: In this article, the authors used the technique of Luttinger surgery to produce small examples of simply connected and non-simply connected minimal symplectic 4-manifolds.
Abstract: In this article we use the technique of Luttinger surgery to produce small examples of simply connected and non-simply connected minimal symplectic 4-manifolds. In particular, we construct: (1) An example of a minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to CP^2#3(-CP^2) which contains a symplectic surface of genus 2, trivial normal bundle, and simply connected complement and a disjoint nullhomologous Lagrangian torus with the fundamental group of the complement generated by one of the loops on the torus. (2) A minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to 3CP^2#5(-CP^2) which has two essential Lagrangian tori with simply connected complement. These manifolds can be used to replace E(1) in many known theorems and constructions. Examples in this article include the smallest known minimal symplectic manifolds with abelian fundamental groups including symplectic manifolds with finite and infinite cyclic fundamental group and Euler characteristic 6.
28 citations
TL;DR: In this article, the error functions of explicit symplectic integrators for solving separable Hamiltonians were exploited to develop explicit time-reversible integrators that are of fractional orders.
Abstract: By exploiting the error functions of explicit symplectic integrators for solving separable Hamiltonians, I show that it is possible to develop explicit time-reversible symplectic integrators for solving nonseparable Hamiltonians of the product form. The algorithms are unusual in that they are of fractional orders.
TL;DR: An important lemma and an induction argument are used to prove the unique solvability, convergence and stability of numerical solutions of a coupled nonlinear Schrodinger system and its convergence is proved.
TL;DR: In this article, the authors derived new relations between the number of focal points of Y i and P i Y i, where P i is an arbitrary symplectic transformation and formulated conditions which guarantee that a given transformation preserves oscillatory properties of transformed systems.
Abstract: This paper studies symplectic transformations for conjoined bases of the symplectic difference systems where the matrix W i is symplectic for We derive new relations between the number of focal points of Y i and P i Y i , where P i is an arbitrary symplectic transformation. We formulate conditions which guarantee that a given transformation preserves oscillatory properties of transformed systems. In particular, for the case and , we establish duality between eventual disconjugacy of general symplectic systems and their reciprocals.
TL;DR: The group of symplectic matrices is explicitly parametrized, and this description is applied to solve two types of problems to describe those matrices that can be certain significant submatrices of a symplectic matrix, and to parametrization of the symp eclectic matrices with a given matrix occurring as a submatrix in a given position.
Abstract: The group of symplectic matrices is explicitly parametrized, and this description is applied to solve two types of problems. First, we describe several sets of structured symplectic matrices, i.e., sets of symplectic matrices that simultaneously have another structure. We consider unitary symplectic matrices, positive definite symplectic matrices, entrywise positive symplectic matrices, totally nonnegative symplectic matrices, and symplectic M-matrices. The special properties of the LU factorization of a symplectic matrix play a key role in the parametrization of these sets. The second class of problems we deal with is to describe those matrices that can be certain significant submatrices of a symplectic matrix, and to parametrize the symplectic matrices with a given matrix occurring as a submatrix in a given position. The results included in this work provide concrete tools for constructing symplectic matrices with special structures or particular submatrices that may be used, for instance, to create examples for testing numerical algorithms.
TL;DR: In this article, a canonical model for the Weil representation of the symplectic group Sp$(V ) was obtained for the case of the Heisenberg group, and a proof of a stronger form of the Stone-von Neumann property for the group was obtained.
Abstract: In this paper, we construct a quantization functor, associating a complex vector space $\cal{H}(V)$ to a finite-dimensional symplectic vector space V over a finite field of odd characteristic. As a result, we obtain a canonical model for the Weil representation of the symplectic group Sp$(V )$. The main new technical result is a proof of a stronger form of the Stone–von Neumann property for the Heisenberg group $H(V )$. Our result answers, for the case of the Heisenberg group, a question of Kazhdan about the possible existence of a canonical vector space attached to a coadjoint orbit of a general unipotent group over finite field.
TL;DR: In this article, a large family of contact manifolds with the following properties was constructed using Lefschetz fibrations: any bounding contact embedding into an exact symplectic manifold satisfying a mild topological assumption is non-displaceable and generically has infinitely many leaf-wise intersection points.
Abstract: We construct using Lefschetz fibrations a large family of contact manifolds with the following properties: Any bounding contact embedding into an exact symplectic manifold satisfying a mild topological assumption is non-displaceable and generically has infinitely many leaf-wise intersection points. Moreover, any Stein filling has infinite dimensional symplectic homology.
TL;DR: In this paper, it was shown that the symplectic reduction of a parabolic contact manifold by a symmetry vector field is special symplectic in a canonical way and that any special manifold or orbifold with such a connection is locally equivalent to one of these symplectic reductions.
Abstract: By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-Kahler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic. We show that the symplectic reduction of (an open cell of) a parabolic contact manifold by a symmetry vector field is special symplectic in a canonical way. Moreover, we show that any special symplectic manifold or orbifold is locally equivalent to one of these symplectic reductions. As a consequence, we are able to prove a number of global properties, including a classification in the compact simply connected case.
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TL;DR: In this paper, Gelfand, Kalinin and Fuks showed that the transverse symplectic class can be decomposed as a product of a certain leaf cohomology class of degree 5.
Abstract: In the early 1970's, Gelfand, Kalinin and Fuks found an exotic characteristic class of degree 7 in the Gelfand-Fuks cohomology of the Lie algebra of formal Hamiltonian vector fields on the plane. We prove that this cohomology class can be decomposed as a product of a certain leaf cohomology class of degree 5 and the transverse symplectic class. This is similar to the well known factorization of the Godbillon-Vey class for codimension n foliations. We also interpret the characteristic classes of transversely symplectic foliations introduced by Kontsevich in terms of the known classes and prove non-triviality for some of them.
TL;DR: It is shown, by considering the method over several steps, that the satisfaction of this condition leads to a reducibility in the method.
Abstract: We derive a criterion that any general linear method must satisfy if it is symplectic. It is shown, by considering the method over several steps, that the satisfaction of this condition leads to a reducibility in the method. Linking the symplectic criterion here to that for Runge–Kutta methods, we demonstrate that a general linear method is symplectic only if it can be reduced to a method with a single input value.
TL;DR: In this article, it was shown that sums of symplectic 4-manifolds along surfaces of positive genus are never rational or ruled, and enumerate each case in which they have Kodaira dimension zero (i.e., are blowups of a manifold with torsion canonical class).
Abstract: Modulo trivial exceptions, we show that symplectic sums of symplectic 4-manifolds along surfaces of positive genus are never rational or ruled, and we enumerate each case in which they have Kodaira dimension zero (i.e., are blowups of symplectic 4-manifolds with torsion canonical class). In particular, a symplectic four-manifold of Kodaira dimension zero arises by such a surgery only if it is diffeomorphic to a blowup either of the K3 surface, the Enriques surface, or a member of a particular family of T2-bundles over T2 each having b1 = 2.
TL;DR: In this paper, the authors studied the behavior of the Poisson bracket under C 0 -pertur-bations and proved that the limit of a converging pseudo-representation of any normed Lie algebra is a representation.
Abstract: The question studied here is the behavior of the Poisson bracket under C 0 -pertur- bations. For this purpose we introduce the notion of pseudo-representation and prove that the limit of a converging pseudo-representation of any normed Lie algebra is a representation. An unexpected consequence of this result is that for many non-closed symplectic manifolds (including cotangent bundles), the group of Hamiltonian diffeomorphisms (with no assumptions on supports) has no C � 1 bi-invariant metric. Our methods also provide a new proof of the Gromov-Eliashberg Theorem, which says that the group of symplectic diffeomorphisms isC 0 - closed in the group of all diffeomorphisms.
TL;DR: A complete characterization of 2 × 2 symplectic matrices with an infinite number of left eigenvalues is given in this paper, using an algorithm for the resolution of equations due to De Leo et al.
Abstract: A complete characterization is obtained of the 2 × 2 symplectic matrices that have an infinite number of left eigenvalues. Also, a new proof is given of a result of Huang and So on the number of eigenvalues of a quaternionic matrix. This is achieved by applying an algorithm for the resolution of equations due to De Leo et al.
TL;DR: In this article, the authors discuss some recent developments concerning the question of when a 4-dimensional ellipsoid can be symplectically embedded in a ball, which turns out to have unexpected relations to the properties of continued fractions and of exceptional curves in blow ups of the complex projective plane.
Abstract: As has been known since the time of Gromov’s Non-squeezing Theorem, symplectic embedding questions lie at the heart of symplectic geometry. After surveying some of the most important ways of measuring the size of a symplectic set, these notes discuss some recent developments concerning the question of when a 4-dimensional ellipsoid can be symplectically embedded in a ball. This problem turns out to have unexpected relations to the properties of continued fractions and of exceptional curves in blow ups of the complex projective plane. It is also related to questions of lattice packing of planar triangles.
TL;DR: In this paper, the concept of symplectic arc length for curves is introduced and the authors construct an adapted symplectic Frenet frame and define 2n - 1 local differential invariants that they call symplectic curvatures of the curve.
Abstract: We consider curves in ℝ2n endowed with the standard symplectic structure. We introduce the concept of symplectic arc length for curves. We construct an adapted symplectic Frenet frame and we define 2n - 1 local differential invariants that we call symplectic curvatures of the curve. We prove that up to a rigid symplectic motion of ℝ2n, there exists a unique curve with prescribed symplectic curvatures. We characterize the curves in ℝ4 with constant symplectic curvatures.
TL;DR: In this paper, the authors use a diffeo-geometric framework based on manifolds that are locally modeled on "convenient" vector spaces to study the geometry of some infinite dimensional spaces.
Abstract: In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on "convenient" vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold (M,!), we construct a weak symplectic structure on each leaf Iw of a foliation of the space of compact oriented isotropic submanifolds in M equipped with top degree forms of total measure 1. These forms are called weightings and such manifolds are said to be weighted. We show that this symplectic structure on the particular leaves consisting of weighted Lagrangian submani- folds is equivalent to a heuristic weak symplectic structure of Weinstein (Adv. Math. 82 (1990), 133-159). When the weightings are positive, these symplectic spaces are symplec- tomorphic to reductions of a weak symplectic structure of Donaldson (Asian J. Math. 3 (1999), 1-15) on the space of embeddings of a fixed compact oriented manifold into M. When M is compact, by generalizing a moment map of Weinstein we construct a symplec- tomorphism of each leaf Iw consisting of positive weighted isotropic submanifolds onto a coadjoint orbit of the group of Hamiltonian symplectomorphisms of M equipped with the Kirillov-Kostant-Souriau symplectic structure. After defining notions of Poisson algebras and Poisson manifolds, we prove that each space Iw can also be identified with a symplectic leaf of a Poisson structure. Finally, we discuss a kinematic description of spaces of weighted submanifolds.
TL;DR: In this article, the authors show that a minimal surface of general type has a canonical symplectic structure (unique up to symplectomorphism) which is invariant for smooth deformation.
Abstract: We show that a minimal surface of general type has a canonical symplectic structure (unique up to symplectomorphism) which is invariant for smooth deformation. We show that the symplectomorphism type is also invariant for deformations which allow certain normal singularities, provided one remains in the same smoothing component. We use this technique to show that the Manetti surfaces yield examples of surfaces of general type which are not deformation equivalent but are canonically symplectomorphic.
TL;DR: In this article, the authors obtained all real solvable and non-nilpotent Lie algebras endowed with a symplectic form that decompose as the direct sum of two ideals.
Abstract: We analyze symplectic forms on six-dimensional real solvable and non-nilpotent Lie algebras. More precisely, we obtain all those algebras endowed with a symplectic form that decompose as the direct sum of two ideals or are indecomposable solvable algebras with a four-dimensional nilradical.
TL;DR: In this paper, the authors investigate situations where this foliation has compact leaves and obtain a space of leaves Y/F which has dimension 2n 2 and admits a holomorphic symplectic form.
Abstract: Let Y be a hypersurface in a 2n-dimensional holomorphic symplectic manifold X. The restriction �|Y of the holomorphic symplectic form induces a rank one foliation on Y . We investigate situations where this foliation has compact leaves; in such cases we obtain a space of leaves Y/F which has dimension 2n 2 and admits a holomorphic symplectic form.
TL;DR: In this article, the authors used twisted Alexander polynomials to study the existence of symplectic structures and the minimal complexity of surfaces in a 4-manifold which admits a free circle action.
Abstract: Let M be a 4-manifold which admits a free circle action. We use twisted Alexander polynomials to study the existence of symplectic structures and the minimal complexity of surfaces in M. The results on the existence of symplectic structures summarize previous results of the authors in \cite{FV08a,FV08,FV07}. The results on surfaces of minimal complexity are new.
TL;DR: In this paper, a numerical method for deriving a symplectic state transition matrix for an arbitrary Hamiltonian dynamical system is presented, which can be applied to accurate, yet computationally efficient, dynamic filters, long-term propagations of the motions of formation-flying spacecraft, eigenstructure/manifold analysis of N-body dynamics.
Abstract: This paper presents a numerical method for deriving a symplectic state transition matrix for an arbitrary Hamiltonian dynamical system. It provides the exact solution-space mapping of the linearized Hamiltonian systems, preserving the symplectic structure that all Hamiltonian systems should possess by nature. The symplectic state transition matrix can be applied to accurate, yet computationally efficient, dynamic filters, long-term propagations of the motions of formation-flying spacecraft, eigenstructure/manifold analysis of N-body dynamics, etc., when the exact structure-preserving property is crucial. We present the derivation and key characteristics of the symplectic state transition matrix and apply it to the two-body dynamics, the circular restricted three-body problem, and an Earth orbit with perturbation forces based on the real ephemeris. These numerical examples reveal that this numerical symplectic state transition matrix shows improvements in preserving the structural properties of the state transition matrix as compared with the conventional linear state transition matrix with Euler or Runge―Kutta integrations.