Showing papers on "Symplectic vector space published in 2004"
TL;DR: In this article, the Calogero-Moser deformation of the orbifold V/G is shown to be a versal Poisson deformation, and the algebra structure on the cohomology H • (X, C ) of any smooth symplectic resolution X↠V/G (multiplicative McKay correspondence) is determined.
205 citations
TL;DR: In this paper, the Von Neumann entropy and mutual information of arbitrary two-mode Gaussian states are derived based on the explicit determination of the symplectic eigenvalues of a generic covariance matrix.
Abstract: We present a derivation of the Von Neumann entropy and mutual information of arbitrary two-mode Gaussian states, based on the explicit determination of the symplectic eigenvalues of a generic covariance matrix. The key role of the symplectic invariants in such a determination is pointed out. We show that the Von Neumann entropy depends on two symplectic invariants, while the purity (or the linear entropy) is determined by only one invariant, so that the two quantities provide two different hierarchies of mixed Gaussian states. A comparison between mutual information and entanglement of formation for symmetric states is considered, taking note of the crucial role of the symplectic eigenvalues in qualifying and quantifying the correlations present in a generic state.
170 citations
TL;DR: In this paper, a nonvanishing result for the invariants of symplectic four-manifolds is established, which leads to new proofs of the indecomposability theorem for symplectic 4-Manifolds.
Abstract: This article analyzes the interplay between symplectic geometry in dimension $4$ and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in [20]. Specifically, we establish a nonvanishing result for the invariants of symplectic four-manifolds, which leads to new proofs of the indecomposability theorem for symplectic four-manifolds and the symplectic Thom conjecture. As a new application, we generalize the indecomposability theorem to splittings of four-manifolds along a certain class of three-manifolds obtained by plumbings of spheres. This leads to restrictions on the topology of Stein fillings of such three-manifolds.
114 citations
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TL;DR: In this paper, it was shown that the bounded derived category of coherent sheaves on a finite-dimensional symplectic vector space is equivalent to the bounded category for coherent coherent sheaving on a subgroup of a finite subgroup.
Abstract: Let $V$ be a finite-dimensional symplectic vector space over a field of characteristic 0, and let $G \subset Sp(V)$ be a finite subgroup. We prove that for any crepant resolution $X \to V/G$, the bounded derived category $D^b(Coh(X))$ of coherent sheaves on $X$ is equivalent to the bounded derived category $D^b_G(Coh(V))$ of $G$-equivariant coherent sheaves on $V$.
91 citations
TL;DR: In this paper, it was shown that a small neighborhood of a closed symplectic submanifold in a geometrically bounded aspherical manifold has non-vanishing symplectic homology.
Abstract: We show that a small neighborhood of a closed symplectic submanifold in a geometrically bounded aspherical symplectic manifold has non-vanishing symplectic homology. As a consequence, we establish the existence of contractible closed characteristics on any thickening of the boundary of the neighborhood. When applied to twisted geodesic flows on compact symplectically aspherical manifolds, this implies the existence of contractible periodic orbits for a dense set of low energy values.
67 citations
TL;DR: In this article, a stack of quantization-deformation modules is constructed on a complex symplectic manifold X, that is, (twisted) modules of microdifferential operators with an extra central parameter τ, a substitute to the lack of homogeneity.
Abstract: On a complex symplectic manifold X, we construct the stack of quantization-deformation modules, that is, (twisted) modules of microdifferential operators with an extra central parameter τ, a substitute to the lack of homogeneity. We also quantize involutive submanifolds of contact manifolds.
63 citations
TL;DR: In this article, the authors studied the existence of left invariant contact forms on Lie groups and proved the uniqueness of this contact form up to a nonzero scalar multiple, which is a necessary and sufficient condition for the presence of a left-invariant contact form on a Lie group.
Abstract: We study left invariant contact forms and left invariant symplectic forms on Lie groups. In the case of filiform Lie groups we give a necessary and sufficient condition for the existence of a left invariant contact form and we prove the uniqueness of this contact form up to a nonzero scalar multiple. As an application we classify all symplectic structures on nilpotent Lie algebras of dimension ⩽6.
57 citations
TL;DR: In this article, a groupoid of triangular bilinear forms was constructed and a relation with the flag variety was established, and the induced Poisson structure and the center of the corresponding algebroid was studied.
Abstract: We construct a symplectic groupoid of triangular bilinear forms and establish a relation with the flag variety. We also study the induced Poisson structure and the centre of the corresponding algebroid.
53 citations
TL;DR: In this paper, a closed 2-form ω is constructed, Poincare dual to α, which is symplectic on the complement of a finite set of unknotted circles Z. The number of circles, counted with sign, is given by d = (c1(s) 2 −3σ(X)−2χ(X))/4, where s is a certain spin C structure naturally associated to ω.
Abstract: Given a smooth, closed, oriented 4-manifold X and α ∈ H2(X, Z) such that α � α > 0, a closed 2-form ω is constructed, Poincare dual to α, which is symplectic on the complement of a finite set of unknotted circles Z. The number of circles, counted with sign, is given by d = (c1(s) 2 −3σ(X)−2χ(X))/4, where s is a certain spin C structure naturally associated to ω.
51 citations
TL;DR: In this paper, a fibration-like structure called a hyperpencil is defined on a smooth, closed 2n-manifold X, generalizing a linear system of curves on an algebraic variety.
Abstract: A fibration-like structure called a hyperpencil is defined on a smooth, closed 2n-manifold X, generalizing a linear system of curves on an algebraic variety. A deformation class of hyperpencils is shown to determine an isotopy class of symplectic structures on X. This provides an inverse to Donaldson's program for constructing linear systems on symplectic manifolds. In dimensions ≤ 6, work of Donaldson and Auroux provides hyperpencils on any symplectic manifold, and the author conjectures that this extends to arbitrary dimensions. In dimensions where this holds, the set of deformation classes of hyperpencils canonically maps onto the set of isotopy classes of rational symplectic forms up to positive scale, topologically determining a dense subset of all symplectic forms up to an equivalence relation on hyperpencils. In particular, the existence of a hyperpencil topologically characterizes those manifolds in dimensions ≤ 6 (and perhaps in general) that admit symplectic structures.
50 citations
TL;DR: In this article, the notion of stabilized fundamental group for the complement of a branch curve in CP 2 was introduced, which generalizes those previously introduced by Moishezon and Teicher for complex projective surfaces.
TL;DR: In this article, the authors describe the corresponding algebra of Weyl-symmetrized functions in operators satisfying nonlinear commutation relations, and the multiplication in this algebra generates an associative product of functions on the phase space.
Abstract: The symplectic geometry of the phase space associated with a charged particle is determined by the addition of the Faraday 2-form to the standard dp ∧ dq structure on In this paper, we describe the corresponding algebra of Weyl-symmetrized functions in operators satisfying nonlinear commutation relations The multiplication in this algebra generates an associative product of functions on the phase space This product is given by an integral kernel whose phase is the symplectic area of a groupoid-consistent membrane A symplectic phase space connection with non-trivial curvature is extracted from the magnetic reflections associated with the Stratonovich quantizer Zero and constant curvature cases are considered as examples The quantization with both static and time-dependent electromagnetic fields is obtained The expansion of the product by the deformation parameter , written in the covariant form, is compared with the known deformation quantization formulae
TL;DR: In this paper, the multiplicity of closed characteristics on partially symmetric convex compact hypersurfaces in R 2n was studied and the main ingredient of the proof is a new (P,ω)-index function and its iteration theory for symplectic matrix paths and some symplectic orthogonal matrix P.
TL;DR: In this paper, it was shown that Wilson loops associated to the natural representation of a structure group G are enough to generate a dense subalgebra of continuous observables on the configuration space of lattice gauge theory with structure groups G. This extends a result of Sengupta [Proc. Am. Math. Soc. Phys. 4 (6) (1980) 515].
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TL;DR: In this article, the classification theory for Lefschetz pencils in the context of symplectic topology is analogous to the correspond- ing theory arising in differential topology.
Abstract: A symplectic structure is canonically constructed on any mani- fold endowed with a topological linear k-system whose fibers carry suitable symplectic data. As a consequence, the classification theory for Lefschetz pencils in the context of symplectic topology is analogous to the correspond- ing theory arising in differential topology. AMS Classification 57R17
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TL;DR: In this paper, the notions of symplectic and orthogonal vector bundles over curves are reviewed, and the connection between principal parts and extensions of vector bundles is discussed, and a criterion for a certain extension of rank 2n to be symplectic or orthogonality is given.
Abstract: We review the notions of symplectic and orthogonal vector bundles over curves, and the connection between principal parts and extensions of vector bundles. We give a criterion for a certain extension of rank 2n to be symplectic or orthogonal. We then describe almost all of its rank n vector subbundles using graphs of sheaf homomorphisms, and give criteria for the isotropy of these subbundles.
TL;DR: In this article, the Hamiltonian dual equations for Reissner plate bending were presented and the transformation from Euclidian space to symplectic space and from Lagrangian system to Hamiltonian system was realized, and the adjoint symplectic orthonormal relation of the eigenfunction vectors for zero eigenvalue were formed.
Abstract: Based on the Hellinger-Reissner variatonal principle for Reissner plate bending and introducing dual variables, Hamiltonian dual equations for Reissner plate bending were presented. Therefore Hamiltonian solution system can also be applied to Reissner plate bending problem, and the transformation from Euclidian space to symplectic space and from Lagrangian system to Hamiltonian system was realized. So in the symplectic space which consists of the original variables and their dual variables, the problem can be solved via effective mathematical physics methods such as the method of separation of variables and eigenfunction-vector expansion. All the eigensolutions and Jordan canonical form eigensolutions for zero eigenvalue of the Hamiltonian operator matrix are solved in detail, and their physical meanings are showed clearly. The adjoint symplectic orthonormal relation of the eigenfunction vectors for zero eigenvalue are formed. It is showed that the all eigensolutions for zero eigenvalue are basic solutions of the Saint-Venant problem and they form a perfect symplectic subspace for zero eigenvalue. And the eigensolutions for nonzero eigenvalue are covered by the Saint-Venant theorem. The symplectic solution method is not the same as the classical semi-inverse method and breaks through the limit of the traditional semi-inverse solution. The symplectic solution method will have vast application.
TL;DR: In this article, the Marsden-Weinstein reduction procedure for almost symplectic manifolds is implemented and conditions for transformation of non-equivariant momentum maps into equivariant ones by modifying the group action are studied.
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TL;DR: In this article, the authors construct a Maslov class which generates the second integral cohomology of Lag(E), and show that its mod 2 reduction is the characteristic class of a flat gerbe with structure group Z2.
Abstract: Let Lag(E) be the Grassmannian of Lagrangian subspaces of a complex symplectic vector space E. We construct a Maslov class which generates the second integral cohomology of Lag(E), and we show that its mod 2 reduction is the characteristic class of a flat gerbe with structure group Z2. We explain the relation of this gerbe to the well-known flat Maslov line bundle with structure group Z4 over the real Lagrangian Grassmannian, whose characteristic class is the mod 4 reduction of the real Maslov class.
TL;DR: In this paper, complex symplectic spaces are used to define algebraic and arithmetic invariants in the theory of symmetric linear operators, which is a generalization of the notion of weak topology.
Abstract: Introduction: motivation and organization of results Complex symplectic spaces: fundamental concepts and definitions Symplectic weak topology Algebraic and arithmetic invariants: Hilbert structures Applications to the theory of symmetric linear operators Aftermath Acknowledgements Bibliography Index.
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TL;DR: In this article, the authors show that the topological terms associated with the inner and outer curvature scalars of the string worldsheet, have a dramatic effect on the global symplectic geometry of the phase space of the theory.
Abstract: Using an imbedding supported background tensor approach for the differential geometry of an imbedded surface in an arbitrary background, we show that the topological terms associated with the inner and outer curvature scalars of the string worldsheet, have a dramatic effect on the global symplectic geometry of the phase space of the theory. By identifying the global symplectic potential of each Lagrangian term in the string action as the argument of the corresponding pure divergence term in a variational principle, we show that those topological terms contribute explicitly to the symplectic potential of any action describing strings, without modifying the string dynamics and the phase space itself. The variation (the exterior derivative on the phase space) of the symplectic potential generates the integral kernel of a covariant and gauge invariant symplectic structure for the theory, changing thus the global symplectic geometry of the phase space. Similar results for non-Abelian gauge theories and General Relativity are briefly discussed
TL;DR: In this article, it was shown that the set of all 2n×2n standard symplectic matrices is closed under multiplication and, thus, forms a semigroup, which can be used recursively to induce an effective structure-preserving algorithm for solving the Riccati equations.
TL;DR: In this article, it was shown that an integrable system over a symplectic manifold whose symplectic form is covariantly constant carries a natural hyper-symplectic structure.
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TL;DR: In this paper, the authors compute the signatures of those symplectic Lefschetz fibra-tions and compute new words in the mapping class group, hence new SFLF signatures.
Abstract: The well-known fact that any genus g symplectic Lefschetz fibra- tion X 4 ! S 2 is given by a word that is equal to the identity element in the mapping class group and each of whose elements is given by a positive Dehn twist, provides an intimate relationship between words in the mapping class group and 4-manifolds that are realized as symplectic Lefschetz fibra- tions. In this article we provide new words in the mapping class group, hence new symplectic Lefschetz fibrations. We also compute the signatures of those symplectic Lefschetz fibrations.
01 Feb 2004
TL;DR: In this paper, an alternative approach to embed second class systems using Wess-Zumino (WZ) variables is proposed, which is developed within the symplectic framework.
Abstract: An alternative approach to embed second class systems using the Wess-Zumino (WZ) variables is proposed[1]. This is developed within the symplectic framework[2,3].
TL;DR: In this paper, a field theory on symplectic manifolds that are fibered over a real surface with interior marked points and cylindrical ends is introduced, and the authors prove a composition theorem in the spirit of QFT and show that this field theory applies naturally to the problem of minimizing geodesics in Hofer's geometry.
Abstract: We introduce in this paper a field theory on symplectic manifolds that are fibered over a real surface with interior marked points and cylindrical ends. We assign to each such object a morphism between certain tensor products of quantum and Floer homologies that are canonically attached to the fibration. We prove a composition theorem in the spirit of QFT, and show that this field theory applies naturally to the problem of minimising geodesics in Hofer’s geometry. This work can be considered as a natural framework that incorporates both the Piunikhin–Salamon–Schwarz morphisms and the Seidel isomorphism.
TL;DR: In this article, it was shown that the spectral sequence of a symplectic fibration collapses at E 2 under certain assumptions on the blowup of CP 5 along a four dimensional submanifold.
Abstract: Moduli spaces of stable pseudoholomorphic curves can be defined parametrically, i.e., over total spaces of symplectic fibrations. This imposes several restrictions on the spectral sequence of a symplectic fibration. We prove, among others, that under certain assumptions the spectral sequence collapses at E2. In the appendix, we prove nontriviality of certain Gromov–Witten invariant for blow-ups. As an application we obtain that any Hamiltonian fibration with the blow-up of CP 5 along four dimensional submanifold as a fibre c-splits. That is its spectral sequence collapses.
TL;DR: In this paper, a simple and natural quantization of differential forms on odd Poisson supermanifolds is given by the relation [f,dg]={f,g} for all functions f and g.
Abstract: There is a simple and natural quantization of differential forms on odd Poisson supermanifolds, given by the relation [f,dg]={f,g} for all functions f and g. We notice that this noncommutative differential algebra has a geometrical realization as a convolution algebra of the symplectic groupoid integrating the Poisson manifold. This quantization is just a part of a quantization of the odd symplectic category (where objects are odd symplectic supermanifolds and morphisms are Lagrangian relations) in terms of 2-graded chain complexes. It is a straightforward consequence of the theory of BV operator acting on semidensities, due to H. Khudaverdian.
Journal Article•
TL;DR: In this article, it was shown that the two-dimensional isotropic harmonic oscillator possesses three independent constants of motion which, with the standard symplectic structure, generates a dynamical symmetry group isomorphic to SU(2).
Abstract: Apart from the total energy, the two-dimensional isotropic harmonic oscillator possesses three independent constants of motion which, with the standard symplectic structure, generates a dynamical symmetry group isomorphic to SU(2). We show that, by suitably redefining the symplectic structure, any of these three constants of motion can be used as a Hamiltonian, and that the remaining two, together with the total energy, generate a dynamical symmetry group isomorphic to SU(1,1). We also show that the standard energy levels of the quantum two-dimensional isotropic harmonic oscillator and their degeneracies are obtained making use of the appropriate representations of SU(1,1), provided that the canonical commutation relations are modified according to the new symplectic structure. Whereas in classical mechanics the different symplectic structures lead to equivalent formulations of the equations of motion, in quantum mechanics the modifications of the commutation relations should be accompanied by modifications in the interpretation of the formalism in order to obtain results equivalent to those found with the common relations.
TL;DR: In this paper, it was shown that any special symplectic connection can be constructed using symplectic realizations of quadratic deformations of a certain linear Poisson structure, and that these Poisson structures cannot be symplectically integrated by a Hausdorff groupoid.
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 any special symplectic connection can be constructed using symplectic realizations of quadratic deformations of a certain linear Poisson structure. Moreover, we show that these Poisson structures cannot be symplectically integrated by a Hausdorff groupoid. As a consequence, we obtain a canonical principal line bundle over any special symplectic manifold or orbifold, and we deduce numerous global consequences.