Showing papers on "Symplectic representation published in 1997"

Symplectic manifolds with no Kähler structure

Abstract: The starting point: Homotopy properties of kahler manifolds.- Nilmanifolds.- Solvmanifolds.- The examples of McDuff.- Symplectic structures in total spaces of bundles.- Survey.

Structure of dynamical systems : a symplectic view of physics

Abstract: The aim of this text is to treat all three basic theories of physics (classical, statistical and quantum mechanics) from the same perspective, namely that of symplectic geometry, in order to show the unifing power of the symplectic geometry approach. The book aims to give the reader an understanding of the interrelationships of the three basic theories of physics. The first two chapters give the necessary mathematical background in differential geometry, Lie groups, and symplectic geometry. In chapter three a symplectic description of Galilean and relativistic mechanics is given, culminating in the classification of elementary particles (relativistic and non-relativistic, with or without spin, with or without mass). In the fourth chapter statistic mechanics is put into symplectic form, finishing with a symplectic description of the kinetic theory of gases and the computation of specific heats. The final chapter covers the author's theory of geometric quantization, and included in this chapter are the derivations of the various wave equations, and the construction of the Fock space. This text is aimed at graduate students and researchers in mathematics and physics who are interested in mathematical and theoretical physics, symplectic geometry, mechanics and geometric quantization.

Asymptotically Holomorphic Families of Symplectic Submanifolds

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.

Disconjugacy and Transformations for Symplectic Systems

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.

Group systems, groupoids, and moduli spaces of parabolic bundles

Abstract: Moduli spaces of homomorphisms or, more generally, twisted homomorphisms from fundamental groups of surfaces to compact connected Lie groups, were connected with geometry through their identification with moduli spaces of holomorphic vector bundles (see [29]). Atiyah and Bott [2] initiated a new approach to the study of these moduli spaces by identifying them with moduli spaces of projectively fiat constant central curvature connections on principal bundles over Riemann surfaces, which they analyzed by methods of gauge theory. In particular, they showed that an invariant inner product on the Lie algebra of the Lie group in question induces a natural symplectic structure on a certain smooth open stratum. Although this moduli space is a finite-dimensional object, generally a stratified space which is locally semialgebraic [19] but sometimes a manifold, its symplectic structure (on the stratum just mentioned) was obtained by applying the method of symplectic reduction to the action of an infinite-dimensional group (the group of gauge transformations) on an infinite-dimensional symplectic manifold (the space of all connections on a principal bundle).

Practical Symplectic Methods with Time Transformation for the Few-Body Problem

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.

Proper group actions and symplectic stratified spaces

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.

A Maslov-type index theory for symplectic paths

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

On Poisson actions of compact Lie groups on symplectic manifolds

Abstract: Let G(P) be a compact simple Poisson-Lie group equipped with a Poisson structure P, and (M, omega) be a symplectic manifold. Assume that M carries a Poisson action of G(P), and there is an equivariant moment map in the sense of Lu and Weinstein which maps

Continuity of Symplectically Adjoint Maps and the Algebraic Structure of Hadamard Vacuum Representations for Quantum Fields on Curved Spacetime

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.

On the Flux Conjectures

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.

Characteristic Classes in Symplectic Topology

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.

Orthosymplectic integration of linear Hamiltonian systems

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.

Examples of quaternionic and Kähler structures in Hamiltonian models of nearly geostrophic flow

Abstract: We study 2-forms on phase spaces of Hamiltonian models of nearly geostrophic flows. A quaternionic structure is identified, and the complex part of a symplectic representation of this structure corresponds to an elliptic Monge - Ampere equation. The real part is an invariant Kahler structure.

Moment maps on symplectic cones

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

Symplectic forms on moduli spaces of flat connections on 2-manifolds

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.

Symplectic Integrator for General Near-Integrable Hamiltonian System

Abstract: In this paper, following the idea of constructing the mixed symplectic integrator (MSI) for a separable Hamiltonian system, we give a low order mixed symplectic integrator for an inseparable, but nearly integrable, Hamiltonian system, Although the difference schemes of the integrators are implicit, they not only have a small truncation error but, due to near integrability, also a faster convergence rate of iterative solution than ordinary implicit integrators, Moreover, these second order integrators are time-reversible.

Gauge theory and symplectic geometry

Abstract: Preface. Participants. Contributors. Lectures on Gauge Theory and Integrable Systems M. Audin. Symplectic Geometry of Plurisubharmonic Functions Y. Eliashberg. Frobenius Manifolds N. Hitchin. Moduli Spaces and Particle Spaces J. Hurtubise. J-Holomorphic Curves and Symplectic Invariants F. Lalonde. Lectures on Gromov Invariants for Symplectic 4-Manifolds D. McDuff. Index.

Basic Properties of Symplectic Dirac Operators

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.

The spinL-function on the symplectic groupGSp(6)

Abstract: In this paper, we will study some essential analytic properties of the “spin”L-function on the symplectic groupGSp (6) (which is associated with the eight-dimensional spin representation of theL-group Gspin (7, ℂ), namely, uniqueness of a bilinear form on an irreducible admissible representation ofGSp (6)×GL(2), local functional equation, and meromorphic continuation, non-vanishing properties at non-archimedean places as well as at archimedean places. Consequently, we will determine the location of the possible poles of the global spinL-function of a generic automorphic cuspidal representation ofGSp(6).

About the supersymmetric extension of the symplectic Faddeev - Jackiw quantization formalism

Abstract: The key equations of the supersymmetric extension of the symplectic Faddeev - Jackiw quantization formalism are written in an alternative way. In this method the crucial problem is to compute the inverse of the symplectic supermatrix. We show how it can be easily given once the configuration space is defined.

Symplectic geometry and the relative Donaldson invariants of

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.