Showing papers on "Symplectic manifold published in 1996"

SW ⇒ Gr: From the Seiberg-Witten equations to pseudo-holomorphic curves

Abstract: The purpose of this article is to explain how pseudo-holomorphic curves in a symplectic 4-manifold can be constructed from solutions to the Seiberg-Witten equations. As such, the main theorem proved here (Theorem 1.3) is an existence theorem for pseudo-holomorphic curves. This article thus provides a proof of roughly half of the main theorem in the announcement [T1]. That theorem, Theorem 4.1, asserts an equivalence between the Seiberg-Witten invariants for a symplectic manifold and a certain Gromov invariant which counts (with signs) the number of pseudoholomorphic curves in a given homology class. The Seiberg-Witten invariants were introduced to mathematicians by Witten [W] based on his joint work with Nat Seiberg [SW1], [SW2]. A description of these invariants is given in Section 1. (See also [KM1], [T1].) Suffice it to say here that when X is a compact, oriented, 4-dimensional manifold with

The symplectic geometry of polygons in Euclidean space

Abstract: We study the symplectic geometry of moduli spaces M r of polygons with xed side lengths in Euclidean space. We show that M r has a natural structure of a complex analytic space and is complex-analytically isomorphic to the weighted quotient of (S 2) n constructed by Deligne and Mostow. We study the Hamiltonian ows on M r obtained by bending the polygon along diagonals and show the group generated by such ows acts transitively on M r. We also relate these ows to the twist ows of Goldman and Jeerey-Weitsman.

The symplectic structure of N=2 supergravity and its central extension

Abstract: We report on the formulation of N = 2, D = 4 supergravity coupled to nv abelian vector multiplets in presence of electric and magnetic charges. General formulae for the (moduli dependent) electric and magnetic charges for the nv + 1 gauge fields are given which reflect the symplectic structure of the underlying special geometry. Model independent sum rules obeyed by these charges are obtained. The specification to Type IIB strings compactified on Calabi-Yau manifolds, with gauge group U(1)h21+1, is given.

Symplectic fibrations and multiplicity diagrams

Abstract: 1. Symplectic fibrations 2. Examples of symplectic fibrations: the coadjoint orbit hierarchy 3. Duistermaat-Heckman polynomials 4. Symplectic fibrations and multiplicity diagrams 5. Computations with orbits Appendices Bibliography Index.

Spectral covers, algebraically completely integrable, hamiltonian systems, and moduli of bundles

Abstract: This is the expanded text of a series of CIME lectures. We present an algebro-geometric approach to integrable systems, starting with those which can be described in terms of spectral curves. The prototype is Hitchin's system on the cotangent bundle of the moduli space of stable bundles on a curve. A variant involving meromorphic Higgs bundles specializes to many familiar systems of mathematics and mechanics, such as the geodesic flow on an ellipsoid and the elliptic solitons. We then describe some systems in which the spectral curve is replaced by various higher dimensional analogues: a spectral cover of an arbitrary variety, a Lagrangian subvariety in an algebraically symplectic manifold, or a Calabi-Yau manifold. One peculiar feature of the CY system is that it is integrable analytically, but not algebraically: the Liouville tori (on which the system is linearized) are the intermediate Jacobians of a family of Calabi-Yau manifolds. Most of the results concerning these three types of non-curve-based systems are quite recent. Some of them, as well as the compatibility between spectral systems and the KP hierarchy, are new, while other parts of the story are scattered through several recent preprints. As best we could, we tried to maintain the survey style of this article, starting with some basic notions in the field and building gradually to the recent developments.

On Riemann-Roch Formulas for Multiplicities

Abstract: A Theorem due to Guillemin and Sternberg about geometric quantization of Hamiltonian actions of compact Lie groups $G$ on compact Kaehler manifolds says that the dimension of the $G$-invariant subspace is equal to the Riemann-Roch number of the symplectically reduced space. Combined with the shifting-trick, this gives explicit formulas for the multiplicities of the various irreducible components. One of the assumptions of the Theorem is that the reduction is regular, so that the reduced space is a smooth symplectic manifold. In this paper, we prove a generalization of this result to the case where the reduced space may have orbifold singularities. Our proof uses localization techniques from equivariant cohomology, and relies in particular on recent work of Jeffrey-Kirwan and Guillemin. Since there are no complex geometry arguments involved, the result also extends to non Kaehlerian settings.

Gromov-Witten invariants of general symplectic manifolds

Abstract: We present an approach to Gromov-Witten invariants that works on arbitrary (closed) symplectic manifolds. We avoid genericity arguments and take into account singular curves in the very formulation. The method is by first endowing mapping spaces from (prestable) algebraic curves into the symplectic manifold with the structure of a Banach orbifold and then exhibiting the space of stable $J$-curves (``stable maps'') as zero set of a Fredholm section of a Banach orbibundle over this space. The invariants are constructed by pairing with a homology class on the locally compact topological space of stable $J$-curves that is generated as localized Euler class of the section.

Symplectic reduction and Riemann-Roch formulas for multiplicities

Abstract: It is well-known that the presence of conserved quantities in a Hamiltonian dynamical system enables one to reduce the number of degrees of freedom of the system. This technique, which goes back to Lagrange and was treated in a modern spirit in papers of Marsden and Weinstein [17] and Meyer [21], is nowadays known as symplectic reduction. In their paper [8] Guillemin and Sternberg considered the problem: what is the quantum analogue of symplectic reduction? In other words, when one quantizes both a mechanical system with symmetries and its reduced system, what is the relationship between the two quantum-mechanical systems that one obtains? Recently a number of authors have made substantial progress in solving this problem, on which I shall report in this note. This development was brought about by work of Witten [27] and subsequent work of Jeffrey and Kirwan [11], Kalkman [13] and Wu [29] on cohomology rings of symplectic quotients. Another important idea turned out to be Lerman’s technique of symplectic cutting or equivariant symplectic surgery [16], a generalization of the notions of blowing up and symplectic reduction.

Symplectic methods for the nonlinear Schrödinger equation

Abstract: In this paper, we show that the spatial discretization of the nonlinear Schrodinger equation leads to a Hamiltonian system, which can be simulated with symplectic numerical schemes. In particular, we apply two symplectic integrators to the nonlinear Schrodinger equation, and we demonstrate that they are able to produce accurate results and to preserve very well the invariants of the original system, such as the energy and charge.

Kahlerian potentials and convexity properties of the moment map

Abstract: Since Kostant proved his convexity theorems for torus actions on ag varieties ([Ko]), there have been numerous contributions to this subject in the more general symplectic and K ahlerian setting. For example, let K be a compact Lie group acting in a Hamiltonian fashion on a connected compact symplectic manifold X . Then the intersection (X )+ of the image of the moment map : X → (LieK)∗ with a positive Weyl chamber t∗ + is convex. Thus (X ) = K · (X )+, where (X )+ is a natural convex section. This was proved by Atiyah and Guillemin–Sternberg ([A], [G-S]) in the abelian case, i.e., where K = T is a compact torus, by Mumford [M]) for X projective algebraic with an integral K ahlerian structure and K not necessarily abelian and in its nal form in the compact symplectic case by Kirwan ([K]). A precise description of (X )+ := (X ) ∩ t∗ + in the compact K ahlerian framework can be given in terms of the image of the components of the set of T xed points in X . In the projective case, where the K ahlerian structure comes from a projective embedding, there are explicit connections to the representations theory of G = KC which are due to Brion and to Mumford–Ness ([B], [N]). The purpose of the present paper is to prove convexity properties of in the setting of non-compact K ahlerian spaces. Note that, by removing an appropriate K-invariant subset from X one can easily construct non-convex -images from convex ones. On the other hand Hilgert–Neeb–Plank proved ([H-N-P]) that convexity holds for a proper moment map : X → (LieK)∗. Another convexity result has been obtained by Sjamaar for X a ne and the restriction of the moment map of a representation ([S2]). Here the representation theoretical approach of Brion is used. Now, let X be an irreducible complex space endowed with a holomorphic action of a complex reductive group G. Assume that a maximal

A convexity theorem for Poisson actions of compact Lie groups

Abstract: When a Poisson Lie group acts on a symplectic manifold, there is a momentum mapping to the dual group. We prove that the image of this momentum mapping meets the positive Weyl chamber in the exponential of a convex set. The result adapts the well-known theorem of Guillemin, Stemberg, and Kirwan to a "multiplicative" (rather than "additive") setting.

Symplectic integration of constrained Hamiltonian systems by composition methods

Abstract: Recent work reported in the literature suggests that for the long-term integration of Hamiltonian dynamical systems one should use methods that preserve the symplectic structure of the flow. In this paper the symplecticity of numerical integrators is investigated for constrained Hamiltonian systems with holonomic constraints. The following two results will be derived. (i) It is shown that any first- or second-order symplectic integrator for unconstrained problems can be generalized to constrained systems such that the resulting scheme is symplectic and preserves the constraints. Based on this, higher-order methods can be derived by the same composition methods used for unconstrained problems.(ii) Leimkuhler and Reich [Math. Comp, 63 (1994), pp. 589–605] derived symplectic integrators based on Dirac’s reformulation of the constrained problem as an unconstrained Hamiltonian system. However, although the unconstrained reformulation can be handled by direct application of any symplectic implicit Runge–Kutta m...

Integration Algorithms and Classical Mechanics

Abstract: Formulation of a new class of fractional-step methods for the incompressible MHD equations that retains the long-term dissipativity of the continuum.. by F. Armero and J. Simo Symplectic methods for conservative multibody systems by E. J. Barth and B. J. Leimkuhler An introduction to symplectic integrators by P. J. Channell and F. R. Neri Symplectic maps and computation of orbits in particle accelerators by A. J. Dragt and D. T. Abell Amold diffusion in symplectic lattice maps by D. J. D. Earn and A. Lichtenberg Integrable Hamiltonians from close approximations to invariant tori by M. Kaasalainen and J. Binney Exhaustive search of symplectic integrators using computer algebra by P. V. Koseleff Conserving algorithms for the $N$ dimensional rigid body by D. K. Lewis and J. Simo More on symplectic correctors by R. I. McLachlan A survey of open problems in symplectic integration by R. I. McLachlan and C. Scovel Symplectic integrators for systems of rigid bodies by S. Reich Backward error analysis of symplectic integrators by J. M. Sanz-Serna Numerical determination of caustics and their bifurcations by T. J. Stuchi and R. Vieira-Martins Symplectic correctors by J. Wisdom, M. Holman, and J. Touma.

Compact Kähler manifolds with hermitian semipositive anticanonical bundle

Abstract: This short note is a continuation of our previous work [DPS93] on compact Kahler manifolds X with semipositive Ricci curvature. Our purpose is to state a splitting theorem describing the structure of such manifolds, and to raise some related questions. The foundational background will be found in papers by Lichnerowicz [Li67], [Li71], and Cheeger-Gromoll [CG71], [CG72]. Recall that a Calabi-Yau manifold X is a compact Kahler manifold with c1(X) = 0 and finite fundamental group π1(X), such that the universal covering X satisfies H0(X,Ω X ) = 0 for all 1 ≤ p ≤ dimX − 1. A symplectic manifold X is a compact Kahler manifold admitting a holomorphic symplectic 2-form ω (of maximal rank everywhere); in particular KX = OX . We denote here as usual

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

Formal deformations of symplectic manifolds with boundary.

Abstract: Suppose that M is a manifold with boundary, supplied with a 6-symplectic structure. For the purpose of this introduction we will mean by that a Poisson structure {,} on C°°(M), non degenerate on vector fields on M whose restriction to dM is in (3 M, TM). An algebra /\\(M) is a formal deformation of (M, {,}) if it is isomorphic äs linear space to C°°(Af )[ft] and has an associative product * which is local, e.g. the jet of /* g at each point depends only on the jets of / and g at that point, and such that the Lie algebra ( ^ , ^]) is a formal deformation of the Lie algebra (C°°(M),{,}). For the precise definition see Section 4. One can think of A* (M) äs the algebra of asymptotic Symbols (in the sense of [9]) of *PDO's constructed out of Hamiltonian vector fields of {,}. The main properties of A(M) are äs follows.

k-integrals and k-Lie symmetries in discrete dynamical systems

Abstract: We generalize the concept of symplectic maps to that of k- symplectic maps: maps whose kth iterates are symplectic. Similarly, k-symmetries and k-integrals are symmetries (resp. integrals) of the kth iterate of the map. It is shown that k-symmetries and k-integrals are related by the k-symplectic structure, as in the k = 1 continuous case (Noether's theorem). Examples are given of k-integrals and their related k-symmetries for k = 1,…,4.

Conjugate points on geodesics of Hofer's metric

Abstract: Hofer's metric on the group of Hamiltonian diffeomorphisms of a symplectic manifold is generated by the L ∞ -norm on the Lie algebra. In this paper we develop a variational theory of geodesics of this metric which satisfy certain non-degeneracy assumptions. We derive the second variation formula, describe conjugate points and obtain necessary and sufficient conditions for the C ∞ -local minimality of such geodesics. We also present an example of a non-degenerate geodesic which is not locally minimal at its first conjugate point.

Squeezing Bogoliubov transformations on the infinite mode CCR‐algebra

Abstract: A detailed analysis of and a general decomposition theorem for in general unbounded symplectic transformations on an arbitrary complex pre‐Hilbert space (one–boson test function space) are given. The structure of strongly continuous symplectic groups on such spaces is determined. The connection between quadratic Hamiltonians, Bogoliubov transformations, and symplectic transformations is discussed in the Fock representation, and their relevance for squeezing operations in quantum optics is pointed out. The results for this rather general class of transformations are proved in a self‐contained fashion.

A Groenewold-van Hove theorem for S**2

Abstract: We prove that there does not exist a nontrivial quantization of the Poisson algebra of the symplectic manifold S2 which is irreducible on the su(2) subalgebra generated by the components {S1, S2, S3} of the spin vector. In fact there does not exist such a quantization of the Poisson subalgebra P consisting of polynomials in {S1, S2, S3}. Furthermore, we show that the maximal Poisson subalgebra of P containing {1, S1, S2, S3} that can be so quantized is just that generated by {1, S1, S2, S3}.

Special geometry and symplectic transformations

Abstract: Special Kahler manifolds are defined by coupling of vector multiplets to $N=2$ supergravity. The coupling in rigid supersymmetry exhibits similar features. These models contain $n$ vectors in rigid supersymmetry and $n+1$ in supergravity, and $n$ complex scalars. Apart from exceptional cases they are defined by a holomorphic function of the scalars. For supergravity this function is homogeneous of second degree in an $(n+1)$-dimensional projective space. Another formulation exists which does not start from this function, but from a symplectic $(2n)$- or $(2n+2)$-dimensional complex space. Symplectic transformations lead either to isometries on the manifold or to symplectic reparametrizations. Finally we touch on the connection with special quaternionic and very special real manifolds, and the classification of homogeneous special manifolds.