Showing papers on "Symplectic vector space published in 1997"
TL;DR: In this article, the Gray map from a corresponding kerdock code to its binary image was shown to induce a geometric map from symplectic to orthogonal spreads, leading to the construction, for any odd composite $m, of large numbers of kerdock codes.
Abstract: When $m$ is odd, spreads in an orthogonal vector space of type $\Omega^+ (2m+2,2)$ are related to binary Kerdock codes and extremal line-sets in $\RR^{2^{m+1}}$ with prescribed angles. Spreads in a $2m$-dimensional binary symplectic vector space are related to Kerdock codes over $\ZZ_4$ and extremal line-sets in $\CC^{2^m}$ with prescribed angles. These connections involve binary, real and complex geometry associated with extraspecial 2-groups. A geometric map from symplectic to orthogonal spreads is shown to induce the Gray map from a corresponding $\ZZ_4$-Kerdock code to its binary image. These geometric considerations lead to the construction, for any odd composite $m$, of large numbers of $\ZZ_4$-Kerdock codes. They also produce new $\ZZ_4$-linear Kerdock and Preparata codes. 1991 Mathematics Subject Classification: primary 94B60; secondary 51M15, 20C99.
336 citations
TL;DR: In this article, the authors derived fourth order symplectic integrators by factorizing the exponential of two operators in terms of an additional higher order composite operator with positive coefficients, and applied these integrators to Kepler's problem.
227 citations
TL;DR: In this paper, a wide range of symplectic submanifolds in a compact symplectic manifold are constructed by tensoring an arbitrary vector bundle by large powers of the complex line bundle whose first Chern class is the symplectic form.
Abstract: We construct a wide range of symplectic submanifolds in a compact symplectic manifold as the zero sets of asymptotically holomorphic sections of vector bundles obtained by tensoring an arbitrary vector bundle by large powers of the complex line bundle whose first Chern class is the symplectic form. We also show that, asymptotically, all sequences of submanifolds constructed from a given vector bundle are isotopic. Furthermore, we prove a result analogous to the Lefschetz hyperplane theorem for the constructed submanifolds.
123 citations
TL;DR: In this paper, the Riccati-type matrix difference equation and a certain quadratic functional play the same role in this theory and their scalar counterparts, and the basic oscillation and transformation properties of symplectic difference systems are established.
Abstract: Basic oscillation and transformation properties of symplectic
difference systems are established. In particular, it is shown
that the Riccati-type matrix difference equation and a certain
quadratic functional play the same role in this theory and
their scalar counterparts.
122 citations
TL;DR: In this article, the use of the extended phase space and time transformations for constructing efficient symplectic algorithms for the investigation of long term behavior of hierarchical few-body systems is discussed, and numerical experiments suggest that the time-transformed generalized leap-frog, combined with symplectic correctors, is one of the most efficient methods for such studies.
Abstract: The use of the extended phase space and time transformations for constructing efficient symplectic algorithms for the investigation of long term behavior of hierarchical few-body systems is discussed. Numerical experiments suggest that the time-transformed generalized leap-frog, combined with symplectic correctors, is one of the most efficient methods for such studies. Applications extend from perturbed two-body motion to hierarchical many-body systems with large eccentricities.
96 citations
TL;DR: In this article, it was shown that if the regularity assumptions are dropped, the reduced space M is a union of symplectic manifolds, and that the symplectic manifold t together in a nice way.
Abstract: Let (M;!) be a Hamiltonian G-space with a momentum map F : M ! g: It is well-known that if is a regular value of F and G acts freely and properly on the level set F 1 (G); then the reduced space M := F 1 (G )=G is a symplectic manifold. We show that if the regularity assumptions are dropped the space M is a union of symplectic manifolds, and that the symplectic manifolds t together in a nice way. In other words the reduced space is a symplectic stratied space. This extends results known for the Hamiltonian action of compact groups.
92 citations
TL;DR: In this article, a topological characterization of the Maslov-type index theory for all continuous degenerate symplectic paths is given, and the basic properties of the index theory are studied.
Abstract: In this paper, we extend the Maslov-type index theory defined in [7], [15], [10], and [18] to all continuous degenerate symplectic paths, give a topological characterization of this index theory for all continuous symplectic paths, and study its basic properties. Suppose τ > 0. We consider an τ -periodic symmetric continuous 2n × 2n matrix function B(t), i.e. B ∈ C(Sτ ,Ls(R)) with Sτ = R/(τZ), L(R2n) being the set of all real 2n×2n matrices, and Ls(R) being the subset of all symmetric matrices. It is well-known that the fundamental solution γ of the linear first order Hamiltonian system
80 citations
TL;DR: In this article, it was shown that for a pair of operators on a symplectic space which are adjoints of each other with respect to the symplectic form (that is, they are sympletically adjoint) that, if they are bounded for some scalar product on the manifold dominating the manifold, then their "purification" induces on the one-particle space of the quantized system a topology which coincides with that given by the two-point functions of quasifree Hadamard states.
Abstract: We derive for a pair of operators on a symplectic space which are adjoints of each other with respect to the symplectic form (that is, they are sympletically adjoint) that, if they are bounded for some scalar product on the symplectic space dominating the symplectic form, then they are bounded with respect to a one-parametric family of scalar products canonically associated with the initially given one, among them being its "purification". As a typical example we consider a scalar field on a globally hyperbolic spacetime governed by the Klein–Gordon equation; the classical system is described by a symplectic space and the temporal evolution by symplectomorphisms (which are symplectically adjoint to their inverses). A natural scalar product is that inducing the classical energy norm, and an application of the above result yields that its "purification" induces on the one-particle space of the quantized system a topology which coincides with that given by the two-point functions of quasifree Hadamard states. These findings will be shown to lead to new results concerning the structure of the local (von Neumann) observable-algebras in representations of quasifree Hadamard states of the Klein–Gordon field in an arbitrary globally hyperbolic spacetime, such as local definiteness, local primarity and Haag-duality (and also split- and type III1-properties). A brief review of this circle of notions, as well as of properties of Hadamard states, forms part of the article.
62 citations
TL;DR: In this paper, the Fredholm determinants of a special class of integral operators K supported on the union of m curve segments in the complex plane are shown to be the tau-functions of an isomonodromic family of meromorphic covariant derivative operators D_l.
Abstract: The Fredholm determinants of a special class of integral operators K supported on the union of m curve segments in the complex plane are shown to be the tau-functions of an isomonodromic family of meromorphic covariant derivative operators D_l. These have regular singular points at the 2m endpoints of the curve segments and a singular point of Poincare index 1 at infinity. The rank r of the vector bundle over the Riemann sphere on which they act equals the number of distinct terms in the exponential sums entering in the numerator of the integral kernels. The deformation equations may be viewed as nonautonomous Hamiltonian systems on an auxiliary symplectic vector space M, whose Poisson quotient, under a parametric family of Hamiltonian group actions, is identified with a Poisson submanifold of the loop algebra Lgl_R(r) with respect to the rational R-matrix structure. The matrix Riemann-Hilbert problem method is used to identify the auxiliary space M with the data defining the integral kernel of the resolvent operator at the endpoints of the curve segments. A second associated isomonodromic family of covariant derivative operators D_z is derived, having rank n=2m, and r finite regular singular points at the values of the exponents defining the kernel of K. This family is similarly embedded into the algebra Lgl_R(n) through a dual parametric family of Poisson quotients of M. The operators D_z are shown to be analogously associated to the integral operator obtained from K through a Fourier-Laplace transform.
57 citations
Posted Content•
TL;DR: For spherically rational manifolds and for those whose minimal Chern number on 2-spheres either vanishes or is large enough, this paper showed that the group of Hamiltonian diffeomorphisms is C^0-closed.
Abstract: The ``Flux conjecture'' for symplectic manifolds states that the group of Hamiltonian diffeomorphisms is C^1-closed in the group of all symplectic diffeomorphisms. We prove the conjecture for spherically rational manifolds and for those whose minimal Chern number on 2-spheres either vanishes or is large enough. We also confirm a natural version of the Flux conjecture for symplectic torus actions. In some cases we can go further and prove that the group of Hamiltonian diffeomorphisms is C^0-closed in the identity component of the group of all symplectic diffeomorphisms.
48 citations
TL;DR: In this article, Katzarkov introduced theories and constructions for cohomology of symplectomorphism groups, including the existence of invariant polynomials in the Lie algebra, the Chern-Simons-type secondary classes, and the symplectic Chern-Weil theory.
Abstract: From the cohomological point of view the symplectomorphism group $Sympl (M)$ of a symplectic manifold is `` tamer'' than the diffeomorphism group. The existence of invariant polynomials in the Lie algebra $\frak {sympl }(M)$, the symplectic Chern-Weil theory, and the existence of Chern-Simons-type secondary classes are first manifestations of this principles. On a deeper level live characteristic classes of symplectic actions in periodic cohomology and symplectic Hodge decompositions. The present paper is called to introduce theories and constructions listed above and to suggest numerous concrete applications. These includes: nonvanishing results for cohomology of symplectomorphism groups (as a topological space, as a topological group and as a discrete group), symplectic rigidity of Chern classes, lower bounds for volumes of Lagrangian isotopies, the subject started by Givental, Kleiner and Oh, new characters for Torelli group and generalizations for automorphism groups of one-relator groups, arithmetic properties of special values of Witten zeta-function and solution of a conjecture of Brylinski. The Appendix, written by L. Katzarkov, deals with fixed point sets of finite group actions in moduli spaces.
TL;DR: In this article, a continuous, orthogonal and symplectic factorization procedure for integrating unstable linear Hamiltonian systems is described, and the method relies on the development of an Orthogonal, symplectic change of variables to block triangular Hamiltonian form.
Abstract: The authors describe a continuous, orthogonal and symplectic factorization procedure for integrating unstable linear Hamiltonian systems. The method relies on the development of an orthogonal, symplectic change of variables to block triangular Hamiltonian form. Integration is thus carried out within the class of linear Hamiltonian systems. Use of an appropriate timestepping strategy ensures that the symplectic pairing of eigenvalues is automatically preserved. For long-term integrations, as are needed in the calculation of Lyapunov exponents, the favorable qualitative properties of such a symplectic framework can be expected to yield improved estimates. The method is illustrated and compared with other techniques in numerical experiments on the Henon-Heiles and spatially discretized Sine-Gordon equations.
Posted Content•
TL;DR: In this article, the authors consider a connected compact Lie group K acting on a symplectic manifold M such that a moment map m exists and determine the Poisson commutant of the algebra of K-invariants.
Abstract: We consider a connected compact Lie group K acting on a symplectic manifold M such that a moment map m exists. A pull-back function via m Poisson commutes with all K-invariants. Guillemin-Sternberg raised the problem to find a converse. In this paper, we solve this problem by determining the Poisson commutant of the algebra of K-invariants. It is completely controlled by the image of m and a certain subquotient W_M of the Weyl group of K. The group W_M is also a reflection group and forms a symplectic analogue of the little Weyl group of a symmetric space. The proof rests ultimately on techniques from algebraic geometry. In fact, a major part of the paper is of independent interest: it establishes connectivity and reducedness properties of the fibers of the (complex algebraic) moment map of a complex cotangent bundle.
TL;DR: In this paper, it was shown that the moment map associated to the torus action of an n-torus on a symplectic cone is a polytopic convex cone in R n.
Abstract: We analyze some convexity properties of the image maps on symplectic cones, similar to the ones obtained by GuilleminSternberg and Atiyah for compact symplectic manifolds in the early 80’s. We prove the image of the moment map associated to the symplectic action of an n-torus on a symplectic cone is a polytopic convex cone in R n : Then, we generalize these results to symplectic manifolds obtained by special perturbations of the symplectic structure of a cone: we obtain sucient (and essentially necessary) conditions for the image of the moment map associated to the perturbed form to remain unchanged. Hamiltonian actions of tori and the images of their moment maps have been intensely studied in the eighties. According to the fundamental result, obtained independently by Atiyah [2] and Guillemin and Sternberg [4], the moment map of a Hamiltonian action of a torus on a compact symplectic manifold has for its image a convex polytope, spanned by the images of the xed points of the action. More recently, Prato [10] proved a convexity result concerning the image of moment maps of torus actions on non-compact symplectic manifolds. Theorem [10]. Let the torus T r act in a Hamiltonian fashion on the symplectic manifold (X;!) and denote by : X !(LieT r ) = R r the corresponding moment map. Suppose that there exists a circle S 1 =fe t 0g T r for some 02 LieT r such that 0 =h; 0iis a proper function having a minimum as its unique critical value. Then (X) is the convex hull of a nite number of rays in (LieT r ). In this paper, we prove a dierent kind of result, closer in spirit to perturbation theory: We start with a special non-compact symplectic manifold, described below, for which a similar convexity theorem holds, and consider the changes of the underlying symplectic structure which keep the image of the resulting moment maps unchanged. Let (X;!) be a symplectic cone with homothety groupft ;t 2 R + gso that t! = t!, for positive t, and compact base X=R + . Suppose that the torus T r acts symplectically on (X;!) and that this action commutes with
Posted Content•
TL;DR: In this article, the authors extend the techniques of Weinstein [W] to give a construction in group cohomology of symplectic forms on twisted moduli spaces of representations of the fundamental group of a 2-manifold.
Abstract: Let $G$ be a compact connected semisimple Lie group. We extend the techniques of Weinstein [W] to give a construction in group cohomology of symplectic forms $\omega$ on \lq twisted' moduli spaces of representations of the fundamental group $\pi$ of a 2-manifold $\Sigma$ (the smooth analogues of ${\rm Hom} (\pi_1(\Sigma), G)/G$) and on relative character varieties of fundamental groups of 2-manifolds. We extend this construction to exhibit a symplectic form on the extended moduli space [J1] (a Hamiltonian $G$-space from which these moduli spaces may be obtained by symplectic reduction), and compute the moment map for the action of $G$ on the extended moduli space.
TL;DR: The Dirac operator on Riemannian manifolds is canonically defined in this article, acting on symplectic spinor fields introduced by B.~Kostant in geometric quantization.
Abstract: Symplectic Dirac operators, acting on symplectic spinor fields introduced by B.~Kostant in geometric quantization, are canonically defined in a similar way as the Dirac operator on Riemannian manifolds. These operators depend on a choice of a metaplectic structure as well as on a choice of a symplectic covariant derivative on the tangent bundle of the underlying manifold. This paper performs a complete study of these relations and shows further basic properties of the symplectic Dirac operators. Various examples are given for illustration.
TL;DR: In this article, the projections of non periodic orbits in a 4-dimensional (4-D) symplectic map composed of two coupled 2-dimensional maps were numerically investigated. But the results were limited to the 2-D maps.
Abstract: We numerically investigate the projections of non periodic orbits in a 4-dimensional (4-D) symplectic map composed of two coupled 2-dimensional (2-D) maps. We describe in detail the structures that are produced in different planes of projection and we find how the morphology of the 4-D orbits is influenced by the features of the 2-D maps as the coupling parameter increases. We give an empirical law that describes this influence.
TL;DR: In this paper, the spin Calogero model was computed in terms of algebro-geometric data on the associated spectral curve, and the symplectic structure of the spin-calogero models was derived.
TL;DR: In this article, a measure and a stochastic Wess-Zumino-Witten Laplacian over the path space of a symplectic manifold with end points in two Lagrangian submanifolds are defined.
TL;DR: In this article, the relative Donaldson invariants of CP are computed directly from the symplectic geometry of the moduli space of framed instantons on CP using Taubes' framework for Donaldson-Floer theory.
Abstract: The purpose of this paper is to present an approach to Computing the relative Donaldson invariants of CP directly from the symplectic geometry of the ^°(CF)y the moduli space of framed instantons on CP We exploit the symplectic geometry of ̂ °(ÜP) to construct an explicit differential form representative for (£)6#( ^°( )) where : #2(CF) s Z[£] -» H(Jf°(CP)) is Donaldson's //-map. We extend this representative to an SO (3)-equivariant class and use Taubes' framework for Donaldson-Floer theory to express the relative Donaldson invariants of ÜP in terms of explicit integrals on the monad construction of the moduli spaces. We prove a general localization result for certain integrals over non-compact symplectic spaces. We apply the result to our integrals on ^°(CP) to further reduce our expression of the invariants to integrals of certain natural differential forme over the jumping divisor in ̂ °(ÜP). We directly compute in the case of instanton number 1. 1991 Mathematics Subject Classification: 14D20; 53C15, 58G.
Posted Content•
TL;DR: In this article, it was shown that there is no finite-dimensional representation by skew-hermitian matrices of a basic algebra of observables B on a noncompact symplectic manifold M for any Lie subalgebra of the Poisson algebra C^\infty(M).
Abstract: We prove that there is no faithful finite-dimensional representation by skew-hermitian matrices of a ``basic algebra of observables'' B on a noncompact symplectic manifold M. Consequently there exists no finite-dimensional quantization of any Lie subalgebra of the Poisson algebra C^\infty(M) containing B.
TL;DR: In this article, it was shown that the eigenvalues of a general positive path can move off the unit circle and that any such path can be extended to have endpoint with all eigen values on the circle.
Abstract: A positive path in the linear symplectic group Sp(2n) is a smooth path which is everywhere tangent to the positive cone. These paths are generated by negative definite (time-dependent) quadratic Hamiltonian functions on Euclidean space. A special case are autonomous positive paths, which are generated by time-independent Hamiltonians, and which all lie in the set u of diagonalizable matrices with eigenvalues on the unit circle. However, as was shown by Krein, the eigenvalues of a general positive path can move off the unit circle. In this paper, we extend Krein’s theory: we investigate the general behavior of positive paths which do not encounter the eigenvalue 1, showing, for example, that any such path can be extended to have endpoint with all eigenvalues on the circle. We also show that in the case 2n = 4 there is a close relation between the index of a positive path and the regions of the symplectic group that such a path can cross. Our motivation for studying these paths came from a geometric squeezing problem [16] in symplectic topology. However, they are also of interest in relation to the stability of periodic Hamiltonian systems [9] and in the theory of geodesies in Riemannian geometry [4].
TL;DR: In this article, the key equations of the Faddeev-Jackiw formalism are written in an alternative way so that the inverse of the symplectic matrix can be easily found, and the nonlinear sigma model including the Hopf term in the action is treated in the framework of this quantization method.
Abstract: The key equations of the symplectic Faddeev-Jackiw formalism are written in an alternative way so that the inverse of the symplectic matrix is easily found. The nonlinear sigma model including the Hopf term in the action is treated in the framework of this quantization method. It is shown how the complete dynamics of the system is described by means of the generalized Faddeev-Jackiw quatum brackets.
18 Mar 1997
TL;DR: For a broad class of manifolds, including symplectic homogeneous spaces (e.g., the sphere), such connection does not exist as mentioned in this paper, which is a consequence of a no-go theorem claiming that the entire Lie algebra of smooth functions on a compact symplectic manifold cannot be quantized, i.e., it has no essentially nontrivial finite-dimensional representations.
Abstract: Dedicated to the memory of Stanis law Zakrzewski Abstract A geometric quantization of a Kahler manifold, viewed as a symplectic manifold, depends on the complex structure compatible with the symplectic form The quantizations form a vector bundle over the space of such complex structures Having a canonical quantization would amount to finding a natural (projectively) flat connection on this vector bundle We prove that for a broad class of manifolds, including symplectic homogeneous spaces (eg, the sphere), such connection does not exist This is a consequence of a "no-go" theorem claiming that the entire Lie algebra of smooth functions on a compact symplectic manifold cannot be quantized, ie, it has no essentially nontrivial finite-dimensional representations
TL;DR: In this article, the authors studied the problem of finding the critical exponent of a setS of non-zero vectors in a non-singular symplectic spaceV of dimension 2m, that is, the minimum of the set {m−dim(U):U∩S=0}.
Abstract: Just as matroids abstract the algebraic properties of determinants in a vector space, Pfaffian structures abstract the algebraic properties of Pfaffians or skew-symmetric determinants in a symplectic space (that is, a vector space with an alternating bilinear form). This is done using an exchange-augmentation axiom which is a combinatorial version of a Laplace expansion or straightening identity for Pfaffians. Using Pfaffian structures, we study a symplectic analogue of the classical critical problem: given a setS of non-zero vectors in a non-singular symplectic spaceV of dimension2m, find its symplectic critical exponent, that is, the minimum of the set {m−dim(U):U∩S=0}, whereU ranges over all the (totally) isotropic subspaces disjoint fromS. In particular, we derive a formula for the number of isotropic subspaces of a given dimension disjoint from the setS by Mobius inversion over the order ideal of isotropic flats in the lattice of flats of the matroid onS given by linear dependence. This formula implies that the symplectic critical exponent ofS depends only on its matroid and Pfaffian structure; however, it may depend on the dimension of the symplectic spaceV.
TL;DR: In this paper, a general method of deriving canonical functions for ray field singularities involving caustics, shadow boundaries and their intersections is presented, where many time-domain canonical functions can be expressed in terms of elementary functions and elliptic integrals.
Abstract: A general method of deriving canonical functions for ray field singularities involving caustics, shadow boundaries and their intersections is presented. It is shown that many time-domain canonical functions can be expressed in terms of elementary functions and elliptic integrals.
TL;DR: In this paper, the notion of harmonic symplectic spinor fields was introduced and the spectrum of the spinor Laplacian on the complex projective space of complex dimension 1.
Abstract: Symplectic spinor fields were considered already in the 70th in order to give the construction of half-densities in the context of geometric quantization. We introduced symplectic Dirac operators acting on symplectic spinor fields and started a systematical investigation. In this paper, we motivate the notion of harmonic symplectic spinor fields. We describe how many linearly independent harmonic symplectic spinors each Riemann surface admits. Furthermore, we calculate the spectrum of the symplectic spinor Laplacian on the complex projective space of complex dimension 1.
TL;DR: In this article, the geometric quantization method with real polarizations is applied to the quantization of a symplectic torus, and it is shown that the Hilbert space constructed through either of these approaches realizes a unitary representation of the integer metaplectic group.
Abstract: We apply the geometric quantization method with real polarizations to the quantization of a symplectic torus. By quantizing with half-densities we canonically associate to the symplectic torus a projective Hilbert space and prove that the projective factor is expressible in terms of the Maslov–Kashiwara index. As in the quantization of a linear symplectic space, we have two ways of resolving the projective ambiguity: (i) by introducing a metaplectic structure and using half-forms in the definition of the Hilbert space; (ii) by choosing a four-fold cover of the Lagrangian Grassmannian of the linear symplectic space covering the torus. We show that the Hilbert space constructed through either of these approaches realizes a unitary representation of the integer metaplectic group.
Posted Content•
TL;DR: In this article, the Pieri-type formula in the cohomology of a Grassmannian of maximal isotropic subspaces of an odd orthogonal or symplectic vector space is proved.
Abstract: We give an elementary proof of the Pieri-type formula in the cohomology of a Grassmannian of maximal isotropic subspaces of an odd orthogonal or symplectic vector space. This proof proceeds by explicitly computing a triple intersection of Schubert varieties. The decisive step is an explicit description of the intersection of two Schubert varieties, from which the multiplicities (which are powers of 2) in the Pieri-type formula are deduced.