Showing papers on "Symplectic vector space published in 1999"

A Class of Symplectic Integrators with Adaptive Time Step for Separable Hamiltonian Systems

Abstract: Symplectic integration algorithms are well suited for long-term integrations of Hamiltonian systems, because they preserve the geometric structure of the Hamiltonian flow. However, this desirable property is generally lost when adaptive time step control is added to a symplectic integrator. We describe an adaptive time step, symplectic integrator that can be used if the Hamiltonian is the sum of kinetic and potential energy components and the required time step depends only on the potential energy (e.g., test-particle integrations in fixed potentials). In particular, we describe an explicit, reversible, symplectic, leapfrog integrator for a test particle in a near-Keplerian potential; this integrator has a time step proportional to distance from the attracting mass and has the remarkable property of integrating orbits in an inverse-square force field with only "along-track" errors; i.e., the phase-space shape of a Keplerian orbit is reproduced exactly, but the orbital period is in error by O(N-2), where N is the number of steps per period.

A stability property of symplectic packing

Abstract: We prove that for any closed symplectic 4-manifold (M,Ω) with [Ω]∈H 2(M, Q) there exists a number N 0 such that for every N≥N 0, (M,Ω) admits full symplectic packing by N equal balls. We also indicate how to compute this N 0. Our approach is based on Donaldson's symplectic submanifold theorem and on tools from the framework of Taubes theory of Gromov invariants.

The Moduli space of complex Lagrangian submanifolds

Abstract: Following an earlier paper on the differential-geometric structure of the moduli space of special Lagrangian submanifolds in a Calabi-Yau manifold, we follow an analogous approach for compact complex Lagrangian submanifolds of a (K\"ahlerian) complex symplectic manifold. The natural geometric structure on the moduli space is a special K\"ahler metric, but we offer a different point of view on the local differential geometry of these, based on the structure of a submanifold of $V\times V$ (where $V$ is a symplectic vector space) which is Lagrangian with respect to two constant symplectic forms. As an application, we show using this point of view how the hyperk\"ahler metric of Cecotti, Ferrara and Girardello associated to a special K\"ahler structure fits into the Legendre transform construction of Lindstr\"om and Ro\v cek.

The moduli space of complex Lagrangian submanifolds

Abstract: Following an earlier paper on the differential-geometric structure of the moduli space of special Lagrangian submanifolds in a Calabi-Yau manifold, we follow an analogous approach for compact complex Lagrangian submanifolds of a (K\"ahlerian) complex symplectic manifold. The natural geometric structure on the moduli space is a special K\"ahler metric, but we offer a different point of view on the local differential geometry of these, based on the structure of a submanifold of $V\times V$ (where $V$ is a symplectic vector space) which is Lagrangian with respect to two constant symplectic forms. As an application, we show using this point of view how the hyperk\"ahler metric of Cecotti, Ferrara and Girardello associated to a special K\"ahler structure fits into the Legendre transform construction of Lindstr\"om and Ro\v cek.

On the Group of Diffeomorphisms Preserving a Locally Conformal Symplectic Structure

Abstract: The automorphism group of a locally conformal symplectic structure is studied. It is shown that this group possesses essential features of the symplectomorphism group. By using a special type of cohomology the flux and Calabi homomorphisms are introduced. The main theorem states that the kernels of these homomorphisms are simple groups (for the precise statement, see Section 7). Some of the methods used may also be interesting in the symplectic case.

The Pfaff lattice and skew-orthogonal polynomials

Abstract: Consider a semi-inflnite skew-symmetric moment matrix, m1 evolving according to the vector flelds @m=@tk = ⁄ k m + m⁄ >k ; where ⁄ is the shift matrix. Then The skew-Borel decomposition m1 := Q i1 JQ >i1 leads to the so-called Pfafi Lattice, which is integrable, by virtue of the AKS theorem, for a splitting involving the a‐ne symplectic algebra. The tau-functions for the system are shown to be pfa‐ans and the wave vectors skew-orthogonal polynomials; we give their explicit form in terms of moments. This system plays an important role in symmetric and symplectic matrix models and in the theory of random matrices (beta=1 or 4).

$4$-manifolds with inequivalent symplectic forms and $3$-manifolds with inequivalent fibrations

Abstract: We exhibit a closed, simply connected 4-manifold X carrying two symplectic structures whose first Chern classes in H(X, Z) lie in disjoint orbits of the diffeomorphism group of X . Consequently, the moduli space of symplectic forms on X is disconnected. The example X is in turn based on a 3-manifold M . The symplectic structures on X come from a pair of fibrations π0,π1 : M → S whose Euler classes lie in disjoint orbits for the action of Diff(M) on H1(M, R).

Symplectic ideals of poisson algebras and the poisson structure associated to quantum matrices

Abstract: (1999). Symplectic ideals of poisson algebras and the poisson structure associated to quantum matrices. Communications in Algebra: Vol. 27, No. 5, pp. 2163-2180.

Schrodinger flows on Grassmannians

Abstract: The geometric non-linear Schrodinger equation (GNLS) on the complex Grassmannian manifold M is the Hamiltonian equation for the energy functional on C(R,M) with respect to the symplectic form induced from the Kahler form on M. It has a Lax pair that is gauge equivalent to the Lax pair of the matrix non-linear Schrodinger equation (MNLS). We construct via gauge transformations an isomorphism from C(R,M) to the phase space of the MNLS equation so that the GNLS flow corresponds to the MNLS flow. The existence of global solutions to the Cauchy problem for GNLS and the hierarchy of commuting flows follows from the correspondence. Direct geometric constructions show the flows are given by geometric partial differential equations, and the space of conservation laws has a structure of a non-abelian Poisson group. We also construct a hierarchy of symplectic structures for GNLS. Under pullback, the known order k symplectic structures correspond to the order (k-2) symplectic structures that we find. The shift by two is a surprise, and is due to the fact that the group structures depend on gauge choice.

Symplectic surfaces in a fixed homology class

Abstract: The purpose of this paper is to investigate the following problem: For a fixed 2-dimensional homology class K in a simply connected symplectic 4-manifold, up to smooth isotopy, how many connected smoothly embedded symplectic submanifolds represent K? We show that when K can be represented by a symplectic torus, there are many instances when K can be representated by infinitely many non-isotopic symplectic tori.

Gromov Invariants and Symplectic Maps

Abstract: Given a symplectomorphism f of a symplectic manifold X, one can form the `symplectic mapping cylinder' $X_f = (X \times R \times S^1)/Z$ where the Z action is generated by $(x,s,t)\mapsto (f(x),s+1,t)$ In this paper we compute the Gromov invariants of the manifolds $X_f$ and of fiber sums of the $X_f$ with other symplectic manifolds This is done by expressing the Gromov invariants in terms of the Lefschetz zeta function of f and, in special cases, in terms of the Alexander polynomials of knots The result is a large set of interesting non-Kahler symplectic manifolds with computational ways of distinguishing them In particular, this gives a simple symplectic construction of the `exotic' elliptic surfaces recently discovered by Fintushel and Stern and of related `exotic' symplectic 6-manifolds

Complex symplectic geometry with applications to ordinary differential operators

Abstract: Complex symplectic spaces, and their Lagrangian subspaces, are defined in accord with motivations from Lagrangian classical dynamics and from linear ordinary differential operators; and then their basic algebraic properties are established. After these purely algebraic developments, an Appendix presents a related new result on the theory of self-adjoint operators in Hilbert spaces, and this provides an important application of the principal theorems. 1. Fundamental definitions for complex symplectic spaces, and three motivating illustrations Complex symplectic spaces, as defined below, are non-trivial generalizations of the real symplectic spaces of Lagrangian classical dynamics [AM], [MA]. Further, these complex spaces provide important algebraic structures clarifying the theory of boundary value problems of linear ordinary differential equations, and the theory of the associated self-adjoint linear operators on Hilbert spaces [AG], [DS], [NA]. These fundamental concepts are introduced in connection with three examples or motivating discussions in this first introductory section, with further technical details and applications presented in the Appendix at the end of this paper. The new algebraic results are given in the second and main section of this paper, which developes the principal theorems of the algebra of finite dimensional complex symplectic spaces and their Lagrangian subspaces. A preliminary treatment of these subjects, with full attention to the theory of self-adjoint operators, can be found in the earlier monograph of these authors [EM]. Definition 1. A complex symplectic space S is a complex linear space, with a prescribed symplectic form [:], namely a sesquilinear form (i) u, v → [u : v], S × S → C, so [c1u + c2v : w] = c1[u : w] + c2[v : w], (1.1) which is skew-Hermitian, (ii) [u : v] = −[v : u], so [u : c1v + c2w] = c̄1[u : v] + c̄2[u : w] Received by the editors August 19, 1997. 1991 Mathematics Subject Classification. Primary 34B05, 34L05; Secondary 47B25, 58F05.

Symplectic geometry and the Verlinde Formulas

Abstract: The purpose of this paper is to give a proof of the Verlinde formulas by applying the Riemann-Roch-Kawasaki theorem to the moduli space of flat G-bundles on a Riemann surface Σ with marked points, when G is a connected simply connected compact Lie group G. Conditions are given for the moduli space to be an orbifold, and the strata are described as moduli spaces for semisimple centralizers in G. The contribution of the strata are evaluated using the formulas of Witten for the symplectic volume, methods of symplectic geometry, including formulas of Witten-Jeffrey-Kirwan, and residue formulas. Our paper extends prior work by Szenes and Jeffrey-Kirwan for SU(n) to general groups G.

An Obstruction to Quantizing Compact Symplectic Manifolds

Abstract: We prove that there are no nontrivial finite-dimensional Lie representations of certain Poisson algebras of polynomials on a compact symplectic manifold. This result is used to establish the existence of a universal obstruction to quantizing a compact symplectic manifold, regardless of the dimensionality of the representation.

Wavelet Approach to Mechanical Problems. Symplectic Group, Symplectic Topology and Symplectic Scales

Abstract: We present the applications of methods from wavelet analysis to polynomial approximations for a number of nonlinear problems. According to the orbit method and by using approach from the geometric quantization theory we construct the symplectic and Poisson structures associated with generalized wavelets by using metaplectic structure. We consider wavelet approach to the calculations of Melnikov functions in the theory of homoclinic chaos in perturbed Hamiltonian systems, for parameterization of Arnold-Weinstein curves in Floer variational approach and characterization of symplectic Hilbert scales of spaces.

On the Darboux theorem for weak symplectic manifolds

Abstract: A new tool to study reducibility of a weak symplectic form to a constant one is introduced and used to prove a version of the Darboux theorem more general than previous ones. More precisely, at each point of the considered manifold a Banach space is associated to the symplectic form (dual of the phase space with respect to the symplectic form), and it is shown that the Darboux theorem holds if such a space is locally constant. The following application is given. Consider a weak symplectic manifold M on which the Darboux theorem is assumed to hold (e.g. a symplectic vector space). It is proved that the Darboux theorem holds also for any finite codimension symplectic submanifolds of M, and for symplectic manifolds obtained from M by the Marsden-Weinstein reduction procedure.

The symplectic geometry of polygons in hyperbolic 3-space

Abstract: We study the symplectic geometry of the moduli spaces Mr = Mr(H 3 ) of closed n-gons with fixed side-lengths in hyperbolic three-space. We prove that these moduli spaces have almost canonical symplectic structures. They are the symplectic quotients of B n by the dressing action of SU(2) (here B is the subgroup of the Borel subgroup of SL2(C) defined below). We show that the hyperbolic Gauss map sets up a real analytic isomorphism between the spaces Mr and the weighted quotients of (S 2 ) n by P SL2(C) studied by Deligne and Mostow. We construct an integrable Hamiltonian system on Mr by bending polygons along nonintersecting diagonals. We describe angle variables and the momentum polyhedron for this system. The results of this paper are the analogues for hyperbolic space of the results of (KM2) for Mr(E 3 ), the space of n-gons with fixed side-lengths in E 3 . We prove Mr(H 3 ) and Mr(E 3 ) are symplectomorphic.

Dynkin diagrams and crepant resolutions of quotient singularities

Abstract: Let $V$ be a complex vector space on which a finite group $G$ acts by linear transformations. Let $W = V \oplus V^*$ be the sum of $V$ with its dual $V^*$. We prove that if the quotient $W/G$ admits a smooth crepant resolution, then the subgroup $G \subset Aut V$ is generated by complex reflections. We also obtain some results on the structure of smooth crepant resolutions of the quotients $W/G$, where $W$ is a symplectic vector space, and $G \subset Aut W$ is a finite group of symplectic linear transformations of the vector space $W$.

A New Integrable Symplectic Map of Neumann Type

Abstract: The nonlinearization approach is generalized to the case of the Neumann constraint associated with a discrete 3×3 matrix eigenvalue problem. A new symplectic map of the Neumann type is obtained by nonlinearization of the discrete eigenvalue problem and its adjoint one. A scheme for generating the involutive system of conserved integrals of the symplectic map is proposed, by which the symplectic map of the Neumann type is further proved to completely integrable. As an application, the calculation of solutions for the hierarchy of lattice soliton equations connected to the discrete eigenvalue problem is reduced to the solutions of a system of ordinary differential equations plus a simple iterative process of the symplectic map of the Neumann type.

On the geography of symplectic 6-manifolds

Abstract: The article investigates the geography of closed, connected and simply connected, six-dimensional manifolds. It is proved that any triple of integers satisfying some necessary arithmetical restrictions occurs as the Chern triple of such a manifold. The main tools used for producing the examples are the symplectic connected sum and the symplectic blow-up.

Exponential sums for symplectic groups and their applications

Abstract: plus another polynomial in q involving certain exponential sums. On the other hand, the expression for (1.2) is similar to that for (1.1), except that the polynomial in q involving (1.3) is multiplied by q − 1 and that the exponential sums appearing in the other polynomial in q are replaced by averages of those exponential sums. In [8], the sums in (1.1) and (1.2) were studied for r = 1 and the connection of the sum in (1.1) with Hodges’ generalized Kloosterman sum over nonsingular alternating matrices was also investigated (cf. [4]–[6]). As the