Showing papers on "Symplectic representation published in 2000"

Introduction to Symplectic Field Theory

Abstract: We sketch in this article a new theory, which we call Symplectic Field Theory or SFT, which provides an approach to Gromov-Witten invariants of symplectic manifolds and their Lagrangian submanifolds in the spirit of topological field theory, and at the same time serves as a rich source of new invariants of contact manifolds and their Legendrian submanifolds. Moreover, we hope that the applications of SFT go far beyond this framework.1

The symplectic Thom conjecture

Abstract: In this paper, we demonstrate a relation among Seiberg-Witten invariants which arises from embedded surfaces in four-manifolds whose self-intersection number is negative. These relations, together with Taubes' basic theorems on the Seiberg-Witten invariants of symplectic manifolds, are then used to prove the symplectic Thom conjecture: a symplectic surface in a symplectic four-manifold is genus-minimizing in its homology class. Another corollary of the relations is a general adjunction inequality for embedded surfaces of negative self-intersection in four-manifolds.

An introduction to symplectic geometry

Abstract: Some aspects of theoretical mechanics Symplectic algebra Symplectic manifolds Hamiltonian vectorfields and the Poisson bracket The moment map Quantization Differentiable manifolds and vector bundles Lie groups and Lie algebras A little cohomology theory Representations of groups Bibliography Index Symbols.

Quantum Spin Dynamics (QSD) : VII. Symplectic Structures and Continuum Lattice Formulations of Gauge Field Theories

Abstract: Interesting non-linear functions on the phase spaces of classical field theories can never be quantized immediately because the basic fields of the theory become operator valued distributions. Therefore, one is usually forced to find a classical substitute for such a function depending on a regulator which is expressed in terms of smeared quantities and which can be quantized in a well-defined way. Namely, the smeared functions define a new symplectic manifold of their own which is easy to quantize. Finally one must remove the regulator and establish that the final operator, if it exists, has the correct classical limit. In this paper we investigate these steps for diffeomorphism invariant quantum field theories of connections. We introduce a generalized projective family of symplectic manifolds, coordinatized by the smeared fields, which is labelled by a pair consisting of a graph and another graph dual to it. We show that there exists a generalized projective sequence of symplectic manifolds whose limit agrees with the symplectic manifold that one started from. This family of symplectic manifolds is easy to quantize and we illustrate the programme outlined above by applying it to the Gauss constraint. The framework developed here is the classical cornerstone on which the semi-classical analysis developed in a new series of papers called ``Gauge Theory Coherent States'' is based.

Symplectic Topology of Integrable Hamiltonian Systems, II: Topological Classification

Abstract: The main purpose of this paper is to give a topological and symplectic classification of completely integrable Hamiltonian systems in terms of characteristic classes and other local and global invariants.

Holomorphic symplectic geometry and orbifold singularities

Abstract: Let G be a finite group acting on a symplectic complex vector space V Assume that the quotient V/G has a holomorphic symplectic resolution We prove that G is generated by "symplectic reflectionsd"', ie symplectomorphisms with fixed space of codimension 2 in V Symplectic resolutions are always semismall A crepant resolution of V/G is always symplectic We give a symplectic version of Nakamura conjectures

Numerical solution of one‐dimensional time‐independent Schrödinger equation by using symplectic schemes

Abstract: The symplectic schemes are extended to the solution of one-dimensional time-independent Schrodinger equation. The Schrodinger equation is first transformed into a Hamiltonian canonical equation by means of the Legendre transformation, and then two methods are developed to solve the numerical solution of the one-dimensional time-independent Schrodinger equation: the symplectic scheme-matrix eigenvalue method and the symplectic scheme-shooting method. Both methods are applied to the calculations of a one-dimensional harmonic oscillator, the hydrogen atom, and a double-well anharmonic oscillator. It is shown that the numerical results of the two methods are nearly the same and are in good agreement with the exact ones when the step length is taken to be properly small. The computation with the symplectic scheme-shooting method spends less computer time than that with the symplectic scheme-matrix eigenvalue method. And thus the symplectic scheme-shooting method is a better numerical method for the calculation of the eigenvalue problem. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 343–349, 2000

Symplectic Methods for the Symplectic Eigenproblem

Abstract: List of Figures. List of Tables. Acknowledgments. 1. Introduction. 2. Preliminaries. 3. The Butterfly Form for Symplectic Matrices and Matrix Pencils. 4. Butterfly SR and SZ Algorithms. 5. The Symplectic Lanczos Algorithm. 6. Concluding Remarks.

An Implicitly Restarted Symplectic Lanczos Method for the Symplectic Eigenvalue Problem

Abstract: An implicitly restarted symplectic Lanczos method for the symplectic eigenvalue problem is presented. The Lanczos vectors are constructed to form a symplectic basis. The inherent numerical difficulties of the symplectic Lanczos method are addressed by inexpensive implicit restarts. The method is used to compute some eigenvalues and eigenvectors of large and sparse symplectic operators.

Symplectic geometry and topology

Abstract: This is a review of the ideas underlying the application of symplectic geometry to Hamiltonian systems. The paper begins with symplectic manifolds and their Lagrangian submanifolds, covers contact manifolds and their Legendrian submanifolds, and indicates the first steps of symplectic and contact topology.

Massey products in symplectic manifolds

Abstract: Massey products in symplectic manifolds are studied. A general method for the construction of symplectic manifolds with non-trivial Massey products of arbitrarily high order is put forward. This method uses symplectic blow-up. The authors find conditions guaranteeing that the symplectic blow-up of X along a submanifold Y inherits non-trivial Massey products from X and Y. As a result, a general construction of non-formal symplectic manifolds by means of symplectic blow-ups is developed.

Nearly Kählerian embeddings of symplectic manifolds

Abstract: Given an integral symplectic manifold, we construct a family of "coherent state" maps into complex projective space. The maps are built from sections of the tensor powers of a hermitian line bundle whose curvature is a multiple of the symplectic form. We show that this family is an almost-complex version of the Kodiara embedding. That is, the maps are embeddings which are approximately pseudo-holomorphic.

Symplectic groups in quantum optics

Abstract: Starting from the symplectic Lie groups Sp(2n,) of an n-mode system in classical and quantum mechanics we discuss, in particular, Sp(2,)~SO(2,1) for a single mode and Sp(4,)~SO(3,2) with some of its important subgroups such as SU(1,1)~Sp(2,) and SU(2). We then look at their applications in quantum optics (squeezing, phase states, beamsplitter and polarization). We explicitly construct the root scheme of the Lie algebra sp(4,) in two different bases as it is an important means for the investigation of the group properties and the construction of the irreducible representations. Apart from the well known realizations of Sp(2,) by single-mode and two-mode systems, we discuss some other realizations of this group by fractal combinations of boson operators.

On nonformal simply-connected symplectic manifolds.

Abstract: For any $N \geq 5$ nonformal simply connected symplectic manifolds of dimension $2N$ are constructed. This disproves the formality conjecture for simply connected symplectic manifolds which was introduced by Lupton and Oprea.

On moduli spaces of symplectic forms

Abstract: We prove that, for any n, there are simply-connected four-manifolds which admit n-tuples of symplectic forms whose first Chern classes have pairwise different divisibilities in integral cohomology. It follows that the moduli space of symplectic forms modulo diffeomorphisms on such a manifold has at least n connected components.

Cover of the brownian bridge and stochastic symplectic action

Abstract: We define the universal cover of the Brownian bridge of a symplectic manifold. This allows us to define a non-trivial functional over it called the stochastic symplectic action, and to define local Sobolev spaces over the universal cover, such that the symplectic action belong to them. This is done in the purpose of an analytical Morse theory in the sense of Witten over the loop space associated to this symplectic action. In this purpose, a stochastic Witten complex is constructed over the universal cover of the loop space, and modulo some weights, it is shown that its cohomology is equal to the stochastic cohomology of the universal cover.

Transversality theory, cobordisms, and invariants of symplectic quotients

Abstract: This paper gives methods for understanding invariants of symplectic quotients. The symplectic quotients considered here are compact symplectic manifolds (or more generally orbifolds), which arise as the symplectic quotients of a symplectic manifold by a compact torus. (A companion paper examines symplectic quotients by a nonabelian group, showing how to reduce to the maximal torus.) Let X be a symplectic manifold, with a Hamiltonian action of a compact torus T. The main topological result of this paper describes an explicit cobordism that exists between a symplectic quotient of X by T, and a collection of iterated projective bundles over components of the set of T-fixed-points. The characteristic classes of these bundles can be determined explicitly, and another result uses this to give formulae for integrals of cohomology classes over the symplectic quotient, in terms of data localized at the T-fixed points of X.

Simple derivation of Symplectic Integrators with First Order Correctors

Abstract: In this paper we consider almost integrable systems for which we show that there is a direct connection between symplectic methods and conventional numerical integration schemes. This enables us to construct several symplectic schemes of varying order. We further show that the symplectic correctors, which formally remove all errors of first order in the perturbation, are directly related to the Euler—McLaurin summation formula. Thus we can construct correctors for these higher order symplectic schemes. Using this formalism we derive the Wisdom—Holman midpoint scheme with corrector and correctors for higher order schemes. We then show that for the same amount of computation we can devise a scheme which is of order O(eh6) + (e2h2), where e is the order of perturbation and h the stepsize. Inclusion of a modified potential further reduces the error to O(eh6) + (e2h4).

On the Siegel-Eisenstein measure and its applications

Abstract: An Eisenstein measure on the symplectic group over rational number field is constructed which interpolatesp-adically the Fourier expansion of Siegel-Eisenstein series. The proof is based on explicit computation of the Fourier expansions by Siegel, Shimura and Feit. As an application of this result ap-adic family of Siegel modular forms is given which interpolates Klingen-Eisenstein series of degree two using Boecherer’s integral representation for the Klingen-Eisenstein series in terms of the Siegel-Eisenstein series.

Exactly solvable quantum spin ladders associated with the orthogonal and symplectic Lie algebras

Abstract: We extend the results of spin ladder models associated with the Lie algebras su(2(n)) to the case of the orthogonal and symplectic algebras o(2(n)), sp(2(n)) where n is the number of legs for the system. Two classes of models are found whose symmetry, either orthogonal or symplectic, has an explicit n dependence. Integrability of these models is shown for an arbitrary coupling of XX-type rung interactions and applied magnetic field term.

The transversal relative equilibria of a Hamiltonian system with symmetry

Abstract: Let P be a symplectic manifold with a free symplectic action of a connected compact Lie group G. We show that, given a certain transversality condition, the set of relative equilibria near pe of a G-invariant Hamiltonian system on P is locally Whitney-stratified by the conjugacy classes of the isotropy subgroups (under the product of the coadjoint and adjoint actions) of the momentum-generator pairs (µ,ξ) of the relative equilibria. The dimension of the stratum of the conjugacy class (K) is dim G + 2dim Z(K)-dim K, where Z(K) is the centre of K. Transverse to this stratum is locally diffeomorphic to the set of commuting pairs of the Lie algebra of K/Z(K). The stratum (K) is a symplectic submanifold of P near pe if and only if pe is non-degenerate and K is a maximal torus of G. We also show that the set of G-invariant Hamiltonians on P for which all the relative equilibria are transversal is open and dense. Thus, generically, the types of singularities of the set of relative equilibria of a Hamiltonian system with symmetry are those types found amongst the singularities at zero of the sets of commuting pairs of certain Lie subalgebras of the symmetry group.

Remarks on the flux groups

Abstract: I study flux groups of compact symplectic manifolds. Under some topological assumptions, I give a new estimate of the rank of flux groups and give a method of construcion of compact symplectic aspherical manifolds.

Normal Forms of Symplectic Matrices

Abstract: In this paper, we prove that for every symplectic matrix M possessing eigenvalues on the unit circle, there exists a symplectic matrix P such that P −1 MP is a symplectic matrix of the normal forms defined in this paper.

Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism

Abstract: To any finite group G of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, H_k, of the smash product of G with the polynomial algebra on V. The algebra H_k, called a symplectic reflection algebra, is related to the coordinate ring of a universal Poisson deformation of the quotient singularity V/G. If G is the Weyl group of a root system in a vector space h and V=h\oplus h^*, then the algebras H_k are `rational' degenerations of Cherednik's double affine Hecke algebra. Let G=S_n, the Weyl group of g=gl_n. We construct a 1-parameter deformation of the Harish-Chandra homomorphism from D(g)^g, the algebra of invariant polynomial differential operators on gl_n, to the algebra of S_n-invariant differential operators with rational coefficients on C^n. The second order Laplacian on g goes, under the deformed homomorphism, to the Calogero-Moser differential operator with rational potential. Our crucial idea is to reinterpret the deformed homomorphism as a homomorphism: D(g)^g \to {spherical subalgebra in H_k}, where H_k is the symplectic reflection algebra associated to S_n. This way, the deformed Harish-Chandra homomorphism becomes nothing but a description of the spherical subalgebra in terms of `quantum' Hamiltonian reduction. In the classical limit k -> \infty, our construction gives an isomorphism between the spherical subalgebra in H_\infty and the coordinate ring of the Calogero-Moser space. We prove that all simple H_\infty-modules have dimension n!, and are parametrised by points of the Calogero-Moser space. The algebra H_\infty is isomorphic to the endomorphism algebra of a distinguished rank n! vector bundle on this space.

Diffeomorphisms, symplectic forms, and Kodaira fibrations

Abstract: As was recently pointed out by McMullen and Taubes [Math Res Lett 6 (1999) 681-696], there are 4-manifolds for which the diffeomorphism group does not act transitively on the deformation classes of orientation-compatible symplectic structures This note points out some other 4-manifolds with this property which arise as the orientation-reversed versions of certain complex surfaces constructed by Kodaira [J Analyse Math 19 (1967) 207-215] While this construction is arguably simpler than that of McMullen and Taubes, its simplicity comes at a price: the examples exhibited herein all have large fundamental groups