Showing papers on "Symplectic vector space published in 2008"
TL;DR: In this article, a collection of vector bundles in the derived categories of coherent sheaves on the Grassmannian of isotropic two-dimensional subspaces in a symplectic vector space of dimension 2n and in an orthogonal vector space for all n was constructed.
Abstract: We construct a full exceptional collection of vector bundles in the derived categories of coherent sheaves on the Grassmannian of isotropic two-dimensional subspaces in a symplectic vector space of dimension 2n and in an orthogonal vector space of dimension 2n + 1 for all n.
109 citations
TL;DR: In this paper, the Calabi-Yau equation on symplectic manifolds was studied and Donaldson's conjecture on estimates for this equation in terms of a taming symplectic form was reduced to an integral estimate of a scalar potential function under a positive curvature condition.
Abstract: We study the Calabi-Yau equation on symplectic manifolds We show that Donaldson's conjecture on estimates for this equation in terms of a taming symplectic form can be reduced to an integral estimate of a scalar potential function Under a positive curvature condition, we show that the conjecture holds
104 citations
TL;DR: In this paper, the authors reveal the topology of co-symplectic/co-Kahler manifolds via symplectic/kahler mapping tori and prove the theorem of Theorem 1.
Abstract: Co-symplectic/co-Kahler manifolds are odd dimensional analog of symplectic/Kahler manifolds, defined early by Libermann in 1959/Blair in 1967 respectively. In this paper, we reveal their topology construction via symplectic/Kahler mapping tori. Namely, Theorem. Co-symplectic manifold = Symplectic mapping torus; Co-Kahler manifold = Kahler mapping torus.
77 citations
TL;DR: In this paper, the Rayleigh principle is used to compare the number of focal points of two conjoined bases of two different configurations of a pair of symplectic difference systems, and it is shown that the numbers differ by at most n.
Abstract: We consider symplectic difference systems together with
associated discrete quadratic functionals and eigenvalue
problems. We establish Sturmian type comparison theorems for
the numbers of focal points of conjoined bases of a pair of
symplectic systems. Then, using this comparison result, we show
that the numbers of focal points of two conjoined bases of one
symplectic system differ by at most n. In the last part of the
paper we prove the Rayleigh principle for symplectic eigenvalue
problems and we show that finite eigenvectors of such
eigenvalue problems form a complete orthogonal basis in the
space of admissible sequences.
60 citations
TL;DR: In this article, the existence of nonformal, simply connected, compact symplectic manifolds of dimension 8 was shown to be true, and the question posed by Babenko and Taimanov was answered in the affirmative.
Abstract: We answer in the affirmative the question posed by Babenko and Taimanov [3] on the existence of nonformal, simply connected, compact symplectic manifolds of dimension 8.
60 citations
TL;DR: In this article, it was shown that a 4-fold of Type A is a double cover of a (singular) sextic hypersurface, and a 4 fold of Type B is birational to a hypersuran of degree at most 12.
Abstract: First steps toward a classification of irreducible symplectic 4-folds whose integral 2-cohomology with 4-tuple cup product is isomorphic to that of (K3)[2]. We prove that any such 4-fold deforms to an irreducible symplectic 4-fold of Type A or Type B. A 4-fold of Type A is a double cover of a (singular) sextic hypersurface and a 4-fold of Type B is birational to a hypersurface of degree at most 12. We conjecture that Type B 4-folds do not exist.
55 citations
01 Jan 2008
TL;DR: In this paper, the connection between closed Newton-Cotes, trigono-metrically-fitted differential methods, symplectic integrators and effi- cient solution of the Schrodinger equation is investigated.
Abstract: In this paper the connection between closed Newton-Cotes, trigono- metrically-fitted differential methods, symplectic integrators and effi- cient solution of the Schrodinger equation is investigated. Several one step symplectic integrators have been obtained based on symplectic ge- ometry, as one can see from the literature. However, the investigation of multistep symplectic integrators is very poor. Zhu et. al. (1) has pre- sented the well known open Newton-Cotes differential methods as mul- tilayer symplectic integrators. The construction of multistep symplectic integrators based on the open Newton-Cotes integration methods was studied by Chiou and Wu (2). In this paper we study the closed Newton- Cotes formulae and we write them as symplectic multilayer structures. We also construct trigonometrically-fitted symplectic methods which are based on the closed Newton-Cotes formulae. We apply the sym- plectic schemes to the well known radial Schrodinger equation in order to investigate the efficiency of the proposed method to these type of problems.
53 citations
TL;DR: In this paper, the formality and the Lefschetz property of ((M, omega) over tilde,(omega) over Tilde) are compared with that of (M, Omega).
48 citations
TL;DR: In this article, the authors combine tools and ideology from two different fields, symplectic geometry and asymptotic geometric analysis, to arrive at some new results, including a dimension-independent bound for the symplectic capacity of a convex body.
Abstract: In this work we bring together tools and ideology from two different fields, symplectic geometry and asymptotic geometric analysis, to arrive at some new results. Our main result is a dimension-independent bound for the symplectic capacity of a convex body ,
44 citations
TL;DR: In this paper, a generalized fixed-point problem was considered from the point of view of some relatively recently discovered symplectic rigidity phenomena, which has interesting applications concerning global perturbations of Hamiltonian systems.
Abstract: In this paper we study a generalized symplectic fixed-point problem, first considered by J. Moser in [20], from the point of view of some relatively recently discovered symplectic rigidity phenomena. This problem has interesting applications concerning global perturbations of Hamiltonian systems. © 2007 Wiley Periodicals, Inc.
44 citations
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TL;DR: In this article, the expected value of the trace of the n-th power of the Frobenius class for an ensemble of hyperelliptic curves of genus g over a fixed finite field in the limit of large genus, and compare the results to the corresponding averages over the unitary symplectic group USp(2g).
Abstract: The zeta function of a curve over a finite field may be expressed in terms of the characteristic polynomial of a unitary symplectic matrix, called the Frobenius class of the curve. We compute the expected value of the trace of the n-th power of the Frobenius class for an ensemble of hyperelliptic curves of genus g over a fixed finite field in the limit of large genus, and compare the results to the corresponding averages over the unitary symplectic group USp(2g). We are able to compute the averages for powers n almost up to 4g, finding agreement with the Random Matrix results except for small n and for n=2g. As an application we compute the one-level density of zeros of the zeta function of the curves, including lower-order terms, for test functions whose Fourier transform is supported in (-2,2). The results confirm in part a conjecture of Katz and Sarnak, that to leading order the low-lying zeros for this ensemble have symplectic statistics.
TL;DR: In this article, the authors define Symplectic cohomology groups for a class of symplectic fibrations with closed symplectic base and convex at infinity fiber, and prove new cases of the Weinstein conjecture.
Abstract: We define Symplectic cohomology groups for a class of symplectic fibrations with closed symplectic base and convex at infinity fiber. The crucial geometric assumption on the fibration is a negativity property reminiscent of negative curvature in complex vector bundles. When the base is symplectically aspherical we construct a spectral sequence of Leray-Serre type converging to the Symplectic cohomology groups of the total space, and we use it to prove new cases of the Weinstein conjecture.
TL;DR: The character of the Weil representation of the metaplectic group Mp(V) was studied in this article, where the final formulas are overtly free of choices (for example, they do not involve the usual choice of a Lagrangian subspace of V).
Abstract: Let V be a symplectic vector space over a finite or local field. We compute the character of the Weil representation of the metaplectic group Mp(V). The final formulas are overtly free of choices (for example, they do not involve the usual choice of a Lagrangian subspace of V). Along the way, in results similar to those of Maktouf, we relate the character to the Weil index of a certain quadratic form, which may be understood as a Maslov index. This relation also expresses the character as the pullback of a certain simple function from Mp(V ⊕ V).
TL;DR: In this paper, the authors constructed the first simply connected minimal symplectic 4-manifold that is homeo-morphic but not diffeomorphic to 3CP 2 #7CP 2.
Abstract: In this article, we construct the first example of a simply-connected minimal symplectic 4-manifold that is homeo- morphic but not diffeomorphic to 3CP 2 #7CP 2 . We also construct the first exotic minimal symplectic CP 2 #5 CP 2 .
TL;DR: In this paper, it was shown that the group of Hamiltonian automorphisms of a symplectic manifold is not finitely many classes of maximal compact tori with respect to the action of the full symplectomorphism group Symp$(M,\omega).
Abstract: We prove that the group of Hamiltonian automorphisms of a symplectic
$4$-manifold $(M,\omega)$, Ham$(M,\omega)$, contains only finitely many
conjugacy classes of maximal compact tori with respect to the action
of the full symplectomorphism group Symp$(M,\omega)$. We also consider
the set of conjugacy classes of\/ $2$-tori in Ham$(M,\omega)$ with
respect to Hamiltonian conjugation and show that its finiteness is
equivalent to the finiteness of the symplectic mapping class group
$\pi_{0}$(Symp$(M,\omega)$). Finally, we extend to rational
and ruled manifolds a result of Kedra which asserts that if $(M,\omega)$ is a simply connected symplectic $4$-manifold with
$b_{2}\geq 3$, and if $(\widetilde{M},\widetilde{\omega}_{\delta})$
denotes a symplectic blow-up of $(M,\omega)$ of small enough capacity
$\delta$, then the rational cohomology algebra of the Hamiltonian
group Ham($\widetilde{M},\widetilde{\omega}_{\delta})$ is not finitely
generated. Our results are based on the fact that in a symplectic
$4$-manifold endowed with any tamed almost structure $J$,
exceptional classes of minimal symplectic area are
$J$-indecomposable.
TL;DR: In this article, the authors introduce blow-up and blowdown operations for generalized complex 4-manifolds and combine these with a surgery analogous to the logarithmic transform, then construct generalized complex structures on nCP2 # m \bar{CP2} for n odd.
Abstract: We introduce blow-up and blow-down operations for generalized complex 4-manifolds. Combining these with a surgery analogous to the logarithmic transform, we then construct generalized complex structures on nCP2 # m \bar{CP2} for n odd, a family of 4-manifolds which admit neither complex nor symplectic structures unless n=1. We also extend the notion of a symplectic elliptic Lefschetz fibration, so that it expresses a generalized complex 4-manifold as a fibration over a two-dimensional manifold with boundary.
TL;DR: In this article, it was shown that S^1 × N does not admit a symplectic structure, where N(P) is the 0-surgery along the pretzel knot P = (5, 3, 5).
Abstract: Let N be a closed, oriented 3-manifold. A folklore conjecture states that S^1 × N admits a symplectic structure only if N admits a fibration over the circle. The purpose of this paper is to provide evidence to this conjecture studying suitable twisted Alexander polynomials of N, and showing that their behavior is the same as of those of fibered 3-manifolds. In particular, we will obtain new obstructions to the existence of symplectic structures and to the existence of symplectic forms representing certain cohomology classes of S^1 × N. As an application of these results we will show that S^1 × N(P) does not admit a symplectic structure, where N(P) is the 0-surgery along the pretzel knot P = (5,-3, 5), answering a question of Peter Kronheimer.
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TL;DR: In this article, the authors exploit the geometric approach to the virtual fundamental class, due to Fukaya-Ono and Li-Tian, to compare stable maps to a symplectic manifold and a submanifold whenever all constrained stable maps contained in the latter to first order.
Abstract: In this paper we exploit the geometric approach to the virtual fundamental class, due to Fukaya-Ono and Li-Tian, to compare the virtual fundamental classes of stable maps to a symplectic manifold and a symplectic submanifold whenever all constrained stable maps to the former are contained in the latter to first order. This extends versions of a statement well-known in the algebraic category to the symplectic category, where it appears to be less familiar. The latter's inherent flexibility then leads to a confirmation of Pandharipande's Gopakumar-Vafa prediction for GW-invariants of Fano classes in 6-dimensional symplectic manifolds. In a forthcoming paper, we use a similar approach to relative Gromov-Witten invariants and the absolute/relative correspondence in genus~0.
TL;DR: In this paper, a quasi-homogeneous variety N is contained in a non-singular Lagrangian submanifold if and only if the algebraic restriction of the symplectic form to N vanishes.
Abstract: We study germs of singular varieties in a symplectic space. In (A1), V. Arnol'd discovered so called ''ghost'' symplectic invariants which are induced purely by singularity. We introduce algebraic restrictions of dierential forms to singular varieties and show that this ghost is exactly the invariants of the algebraic restriction of the sym- plectic form. This follows from our generalization of Darboux-Givental' theorem from non-singular submanifolds to arbitrary quasi-homogeneous varieties in a symplectic space. Using algebraic restrictions we introduce new symplectic invariants and explain their geo- metric meaning. We prove that a quasi-homogeneous variety N is contained in a non- singular Lagrangian submanifold if and only if the algebraic restriction of the symplectic form to N vanishes. The method of algebraic restriction is a powerful tool for various classification problems in a symplectic space. We illustrate this by complete solutions of symplectic classification problem for the classical A, D, E singularities of curves, the S5 sin- gularity, and for regular union singularities.
TL;DR: In this article, the authors studied the structure of quandles with an antisymmetric bilinear form and showed that every finite dimensional quandle over a finite field of characteristic other than 2 is a disjoint union of a trivial quandler and a connected quander.
Abstract: We study the structure of symplectic quandles, quandles which are also $R$-modules equipped with an antisymmetric bilinear form. We show that every finite dimensional symplectic quandle over a finite field $\mathbb{F}$ or arbitrary field $\mathbb{F}$ of characteristic other than 2 is a disjoint union of a trivial quandle and a connected quandle. We use the module structure of a symplectic quandle over a finite ring to refine and strengthen the quandle counting invariant.
TL;DR: In this article, the role of the extended Clifford group in classifying the spectra of phase point operators within the framework laid out by Gibbs et al. for setting up Wigner distributions on discrete phase spaces based on finite fields was analyzed.
Abstract: We analyze the role of the extended Clifford group in classifying the spectra of phase point operators within the framework laid out by [Gibbons et al., Phys. Rev. A 70, 062101 (2004)] for setting up Wigner distributions on discrete phase spaces based on finite fields. To do so we regard the set of all the discrete phase spaces as a symplectic vector space over the finite field. Auxiliary results include a derivation of the conjugacy classes of ESL(2,FN).
TL;DR: In this paper, the authors give an overview of Donaldson's theory of linear systems on symplectic manifolds and the algebraic and geometric invariants to which they give rise, and discuss invariants obtained by combining this theory with pseudo-holomorphic curve methods.
Abstract: This set of lectures aims to give an overview of Donaldson's theory of linear systems on symplectic manifolds and the algebraic and geometric invariants to which they give rise. After collecting some of the relevant background, we discuss topological, algebraic and symplectic viewpoints on Lefschetz pencils and branched covers of the projective plane. The later lectures discuss invariants obtained by combining this theory with pseudo-holomorphic curve methods.
TL;DR: In this paper, it was shown that the real symplectic form Re Ω (respectively, Im Ω) on O is exact if and only if all the eigenvalues ad( c ) are real.
01 Jan 2008
TL;DR: In this article, the authors define Dirac-isospectral noncommutative deformations of the spectral triples of locally anti de Sitter black holes, based on Universal Deformation Quantization Quantization Formulas (UDF) obtained from an oscillatory integral kernel on an appropriate symmetric space.
Abstract: We realize quantized anti de Sitter space black holes, building Connes spectral triples, similar to those used for quantized spheres but based on Universal Deformation Quantization Formulas (UDF) obtained from an oscillatory integral kernel on an appropriate symplectic symmetric space. More precisely we first obtain a UDF for Lie subgroups acting on a symplectic symmetric space M in a locally simply transitive manner. Then, observing that a curvature contraction canonically relates anti de Sitter geometry to the geometry of symplectic symmetric spaces, we use that UDF to define what we call Dirac-isospectral noncommutative deformations of the spectral triples of locally anti de Sitter black holes. The study is motivated by physical and cosmological considerations. Comment: 24 pages, to appear in Contemporary Mathematics (AMS) in the volume of the proceedings of the conference Poisson 2006 held at Keio Univ (Japan)
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TL;DR: In this paper, a nontrivial cyclic cocycle on the Weyl algebra of a symplectic vector space is constructed, and a quasi-isomorphism from the complex of differential forms on a manifold to the cyclic cochains of any formal deformation quantization thereof is derived.
Abstract: We construct a nontrivial cyclic cocycle on the Weyl algebra of a symplectic vector space. Using this cyclic cocycle we construct an explicit, local, quasi-isomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof. We give a new proof of Nest-Tsygan's algebraic higher index theorem by computing the pairing between such cyclic cocycles and the $K$-theory of the formal deformation quantization. Furthermore, we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold. As an application, we obtain the analytic higher index theorem of Connes--Moscovici and its extension to orbifolds.
TL;DR: This article showed that Poisson fibrations integrate to a special kind of groupoids, called fibered symplectic groupoids (FPSG), and showed that these groupoids integrate with Poisson Fibrations.
Abstract: We show that Poisson fibrations integrate to a special kind of symplectic fibrations, called fibered symplectic groupoids.
TL;DR: Using invariance by fixed-endpoints homotopies and a generalized notion of symplectic Cayley transform, this paper proved a product formula for the Conley-Zehnder index of continuous paths with arbitrary endpoints in the symplectic group.
Abstract: Using invariance by fixed-endpoints homotopies and a generalized notion of symplectic Cayley transform, we prove a product formula for the Conley–Zehnder index of continuous paths with arbitrary endpoints in the symplectic group. We discuss two applications of the formula, to the metaplectic group and to periodic solutions of Hamiltonian systems.
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TL;DR: The Schroedinger equation is a lift of Newton's law of motion on the space of probability measures, where derivatives are taken wrt the Wasserstein Riemannian metric as mentioned in this paper, where the potential is the sum of the total classical potential energy of the extended system and its Fisher information.
Abstract: We show that the Schroedinger equation is a lift of Newton's law of motion on the space of probability measures, where derivatives are taken wrt the Wasserstein Riemannian metric Here the potential is the sum of the total classical potential energy of the extended system and its Fisher information The precise relation is established via a well known ('Madelung') transform which is shown to be a symplectic submersion of the standard symplectic structure of complex valued functions into the canonical symplectic space over the Wasserstein space All computations are conducted in the framework of Otto's formal Riemannian calculus for optimal transportation of probability measures
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TL;DR: In this paper, it was shown that every degree 2 homology class of a 2n-dimensional manifold is represented by an immersed symplectic surface if it has positive symplectic area.
Abstract: In this paper we show that every degree 2 homology class of a 2n-dimensional symplectic manifold is represented by an immersed symplectic surface if it has positive symplectic area Moreover, the sym- plectic surface can be chosen to be embedded if 2n is at least 6 We also analyze the additional conditions under which embedded symplectic rep- resentatives exist in dimension 4