Showing papers on "Symplectic manifold published in 2008"

Variational Symplectic Integrator for Long-Time Simulations of the Guiding-Center Motion of Charged Particles in General Magnetic Fields

Abstract: A variational symplectic integrator for the guiding-center motion of charged particles in general magnetic fields is developed for long-time simulation studies of magnetized plasmas. Instead of discretizing the differential equations of the guiding-center motion, the action of the guiding-center motion is discretized and minimized to obtain the iteration rules for advancing the dynamics. The variational symplectic integrator conserves exactly a discrete Lagrangian symplectic structure, and has better numerical properties over long integration time, compared with standard integrators, such as the standard and variable time-step fourth order Runge-Kutta methods.

Rigidity and uniruling for Lagrangian submanifolds

Abstract: This paper explores the topology of monotone Lagrangian submanifolds $L$ inside a symplectic manifold $M$ by exploiting the relationships between the quantum homology of $M$ and various quantum structures associated to the Lagrangian $L$.

Taming symplectic forms and the Calabi-Yau equation

Abstract: We study the Calabi-Yau equation on symplectic manifolds We show that Donaldson's conjecture on estimates for this equation in terms of a taming symplectic form can be reduced to an integral estimate of a scalar potential function Under a positive curvature condition, we show that the conjecture holds

A Groupoid Approach to Quantization

Abstract: Many interesting $C∗$-algebras can be viewed as quantizations of Poisson manifolds. I propose that a Poisson manifold may be quantized by a twisted polarized convolution $C∗$-algebra of a symplectic groupoid. Toward this end, I define polarizations for Lie groupoids and sketch the construction of this algebra. A large number of examples show that this idea unifies previous geometric constructions, including geometric quantization of symplectic manifolds and the $C∗$-algebra of a Lie groupoid. I sketch a few new examples, including twisted groupoid $C∗$-algebras as quantizations of bundle affine Poisson structures.

Rational symplectic field theory over Z2 for exact Lagrangian cobordisms

Abstract: We construct a version of rational Symplectic Field Theory for pairs $(X,L)$, where $X$ is an exact symplectic manifold, where $L\subset X$ is an exact Lagrangian submanifold with components subdivided into $k$ subsets, and where both $X$ and $L$ have cylindrical ends. The theory associates to $(X,L)$ a $\Z$-graded chain complex of vector spaces over $\Z_2$, filtered with $k$ filtration levels. The corresponding $k$-level spectral sequence is invariant under deformations of $(X,L)$ and has the following property: if $(X,L)$ is obtained by joining a negative end of a pair $(X',L')$ to a positive end of a pair $(X'',L'')$, then there are natural morphisms from the spectral sequences of $(X',L')$ and of $(X'',L'')$ to the spectral sequence of $(X,L)$. As an application, we show that if $\Lambda\subset Y$ is a Legendrian submanifold of a contact manifold then the spectral sequences associated to $(Y\times\R,\Lambda_k^s\times\R)$, where $Y\times\R$ is the symplectization of $Y$ and where $\Lambda_k^s\subset Y$ is the Legendrian submanifold consisting of $s$ parallel copies of $\Lambda$ subdivided into $k$ subsets, give Legendrian isotopy invariants of $\Lambda$.

Topology of Co-symplectic/Co-Kähler Manifolds

Abstract: Co-symplectic/co-Kahler manifolds are odd dimensional analog of symplectic/Kahler manifolds, defined early by Libermann in 1959/Blair in 1967 respectively. In this paper, we reveal their topology construction via symplectic/Kahler mapping tori. Namely, Theorem. Co-symplectic manifold = Symplectic mapping torus; Co-Kahler manifold = Kahler mapping torus.

Base manifolds for fibrations of projective irreducible symplectic manifolds

Abstract: Given a projective irreducible symplectic manifold M of dimension 2n, a projective manifold X and a surjective holomorphic map f:M→X with connected fibers of positive dimension, we prove that X is biholomorphic to the projective space of dimension n. The proof is obtained by exploiting two geometric structures at general points of X: the affine structure arising from the action variables of the Lagrangian fibration f and the structure defined by the variety of minimal rational tangents on the Fano manifold X.

Toeplitz operators on symplectic manifolds

Abstract: We study the Berezin-Toeplitz quantization on symplectic manifolds making use of the full off-diagonal asymptotic expansion of the Bergman kernel. We give also a characterization of Toeplitz operators in terms of their asymptotic expansion. The semi-classical limit properties of the Berezin-Toeplitz quantization for non-compact manifolds and orbifolds are also established.

Nekhoroshev stability at L4 or L5 in the elliptic‐restricted three‐body problem – application to Trojan asteroids

Abstract: The problem of analytical determination of the stability domain around the points L 4 or L 5 of the Lagrangian equilateral configuration of the three-body problem has served in the literature as a basic celestial mechanical model probing the predictive power of the so-called Nekhoroshev theory of exponential stability in non-linear Hamiltonian dynamical systems. All analytical investigations in this framework have so far been based on the circular restricted three-body problem (CRTBP). In this work, we extend the analytical estimates of Nekhoroshev stability in the case of the planar elliptic-restricted three-body problem (ERTBP). To this end, we introduce an explicit symplectic mapping model for the planar ERTBP, obtained via Hadjidemetriou's method, which generalizes the family of mappings discussed in earlier papers. The mapping is based on an expansion of the disturbing function up to a sufficiently high order in the eccentricities and the variations of the semimajor axis, and it is given as a series around a period-one fixed point of the system. Within the domain of the mapping's convergence, we then compute a Birkhoff normal form for 4D symplectic mappings as well as the associated approximate integrals of motion which can be expressed in terms of proper elements. The variations of the integrals predicted by the remainder of the normal form series provide a lower bound for the domain of Nekhoroshev stability for a time at least equal to the age of the Solar System. In the case of Jupiter's Trojans, the domain of the mapping's convergence lies entirely within the region of librational motion, in which the longitude of the perihelion of the asteroid librates around a fixed point value of ϖ. For most asteroids outside this domain macroscopic chaotic diffusion cannot be ruled out. The present analysis provides a physically relevant estimate of the domain of Nekhoroshev stability for proper librations (Dp < 10°), and marginal for proper eccentricities (e p < 0.01). The formalism is developed in a general way allowing for applications in both, our Solar System and extrasolar system dynamics.

Immersed Lagrangian Floer Theory

Abstract: Let (M,w) be a compact symplectic manifold, and L a compact, embedded Lagrangian submanifold in M. Fukaya, Oh, Ohta and Ono construct Lagrangian Floer cohomology for such M,L, yielding groups HF^*(L,b;\Lambda) for one Lagrangian or HF^*((L,b),(L',b');\Lambda) for two, where b,b' are choices of bounding cochains, and exist if and only if L,L' have unobstructed Floer cohomology. These are independent of choices up to canonical isomorphism, and have important invariance properties under Hamiltonian equivalence. Floer cohomology groups are the morphism groups in the derived Fukaya category of (M,w), and so are an essential part of the Homological Mirror Symmetry Conjecture of Kontsevich. The goal of this paper is to extend all this to immersed Lagrangians L in M with immersion i : L --> M, with transverse self-intersections. In the embedded case, Floer cohomology HF^*(L,b;\Lambda) is a modified, 'quantized' version of cohomology H^*(L;\Lambda) over the Novikov ring \Lambda. In our immersed case, HF^*(L,b;\Lambda) turns out to be a quantized version of the sum of H^*(L;\Lambda) with a \Lambda-module spanned by pairs (p,q) for p,q distinct points of L with i(p)=i(q) in M. The theory becomes simpler and more powerful for graded Lagrangians in Calabi-Yau manifolds, when we can work over a smaller Novikov ring \Lambda_{CY}. The proofs involve associating a gapped filtered A-infinity algebra over \Lambda or \Lambda_{CY} to i : L --> M, which is independent of nearly all choices up to canonical homotopy equivalence, and is built using a series of finite approximations called A_{N,0} algebras for N=0,1,2,...

Sturmian and spectral theory for discrete symplectic systems

Abstract: We consider symplectic difference systems together with associated discrete quadratic functionals and eigenvalue problems. We establish Sturmian type comparison theorems for the numbers of focal points of conjoined bases of a pair of symplectic systems. Then, using this comparison result, we show that the numbers of focal points of two conjoined bases of one symplectic system differ by at most n. In the last part of the paper we prove the Rayleigh principle for symplectic eigenvalue problems and we show that finite eigenvectors of such eigenvalue problems form a complete orthogonal basis in the space of admissible sequences.

An 8-dimensional nonformal, simply connected, symplectic manifold

Abstract: We answer in the affirmative the question posed by Babenko and Taimanov [3] on the existence of nonformal, simply connected, compact symplectic manifolds of dimension 8.

Closed Newton–Cotes trigonometrically-fitted formulae of high order for the numerical integration of the Schrödinger equation

Abstract: In this paper we investigate the connection between (i) closed Newton–Cotes formulae, (ii) trigonometrically-fitted differential methods, (iii) symplectic integrators and (iv) efficient solution of the Schrodinger equation. In the last decades several one step symplectic integrators have been produced based on symplectic geometry, (see the relevant literature and the references here). However, the study of multistep symplectic integrators is very poor. In this paper we investigate the closed Newton–Cotes formulae and we write them as symplectic multilayer structures. We also develop trigonometrically-fitted symplectic methods which are based on the closed Newton–Cotes formulae. We apply the symplectic schemes to the well known radial Schrodinger equation in order to investigate the efficiency of the proposed method to these type of problems.

Irreducible symplectic 4-folds numerically equivalent to (k3)[2]

Abstract: First steps toward a classification of irreducible symplectic 4-folds whose integral 2-cohomology with 4-tuple cup product is isomorphic to that of (K3)[2]. We prove that any such 4-fold deforms to an irreducible symplectic 4-fold of Type A or Type B. A 4-fold of Type A is a double cover of a (singular) sextic hypersurface and a 4-fold of Type B is birational to a hypersurface of degree at most 12. We conjecture that Type B 4-folds do not exist.

Branes and Quantization

Abstract: The problem of quantizing a symplectic manifold (M,\omega) can be formulated in terms of the A-model of a complexification of M. This leads to an interesting new perspective on quantization. From this point of view, the Hilbert space obtained by quantization of (M,\omega) is the space of (Bcc,B') strings, where Bcc and B' are two A-branes; B' is an ordinary Lagrangian A-brane, and Bcc is a space-filling coisotropic A-brane. B' is supported on M, and the choice of \omega is encoded in the choice of Bcc. As an example, we describe from this point of view the representations of the group SL(2,R). Another application is to Chern-Simons gauge theory.

Closed Newton-Cotes Trigonometrically-Fitted Formulae for the Solution of the Schrodinger Equation

Abstract: In this paper the connection between closed Newton-Cotes, trigono- metrically-fitted differential methods, symplectic integrators and effi- cient solution of the Schrodinger equation is investigated. Several one step symplectic integrators have been obtained based on symplectic ge- ometry, as one can see from the literature. However, the investigation of multistep symplectic integrators is very poor. Zhu et. al. (1) has pre- sented the well known open Newton-Cotes differential methods as mul- tilayer symplectic integrators. The construction of multistep symplectic integrators based on the open Newton-Cotes integration methods was studied by Chiou and Wu (2). In this paper we study the closed Newton- Cotes formulae and we write them as symplectic multilayer structures. We also construct trigonometrically-fitted symplectic methods which are based on the closed Newton-Cotes formulae. We apply the sym- plectic schemes to the well known radial Schrodinger equation in order to investigate the efficiency of the proposed method to these type of problems.

Constructing Infinitely Many Smooth Structures on Small 4-Manifolds

Abstract: The purpose of this article is twofold. First we outline a general construction scheme for producing simply-connected minimal symplectic 4-manifolds with small Euler characteristics. Using this scheme, we illustrate how to obtain irreducible symplectic 4-manifolds homeomorphic but not diffeomorphic to $\CP#(2k+1)\CPb$ for $k = 1,...,4$, or to $3\CP# (2l+3)\CPb$ for $l =1,...,6$. Secondly, for each of these homeomorphism types, we show how to produce an infinite family of pairwise nondiffeomorphic nonsymplectic 4-manifolds belonging to it. In particular, we prove that there are infinitely many exotic irreducible nonsymplectic smooth structures on $\CP#3\CPb$, $3\CP#5\CPb$ and $3\CP#7\CPb$.

Spectral invariants in Lagrangian Floer theory

Abstract: Let $(M,\omega)$ be a symplectic manifold compact or convex at infinity. Consider a closed Lagrangian submanifold $L$ such that $\omega |_{\pi_2(M,L)}=0$ and $\mu|_{\pi_2(M,L)}=0$, where $\mu$ is the Maslov index. Given any Lagrangian submanifold $L'$, Hamiltonian isotopic to $L$, we define Lagrangian spectral invariants associated to the non zero homology classes of $L$, depending on $L$ and $L'$. We show that they naturally generalize the Hamiltonian spectral invariants introduced by Oh and Schwarz, and that they are the homological counterparts of higher order invariants, which we also introduce here, via spectral sequence machinery introduced by Barraud and Cornea. These higher order invariants are new even in the Hamiltonian case and carry strictly more information than the classical ones. We provide a way to distinguish them one from another and estimate their difference in terms of a geometric quantity.

The "M"-ellipsoid, symplectic capacities and volume

Abstract: In this work we bring together tools and ideology from two different fields, symplectic geometry and asymptotic geometric analysis, to arrive at some new results. Our main result is a dimension-independent bound for the symplectic capacity of a convex body ,

Symplectic Rigidity, Symplectic Fixed Points, and Global Perturbations of Hamiltonian Systems

Abstract: In this paper we study a generalized symplectic fixed-point problem, first considered by J. Moser in [20], from the point of view of some relatively recently discovered symplectic rigidity phenomena. This problem has interesting applications concerning global perturbations of Hamiltonian systems. © 2007 Wiley Periodicals, Inc.

Traces of high powers of the Frobenius class in the hyperelliptic ensemble

Abstract: The zeta function of a curve over a finite field may be expressed in terms of the characteristic polynomial of a unitary symplectic matrix, called the Frobenius class of the curve. We compute the expected value of the trace of the n-th power of the Frobenius class for an ensemble of hyperelliptic curves of genus g over a fixed finite field in the limit of large genus, and compare the results to the corresponding averages over the unitary symplectic group USp(2g). We are able to compute the averages for powers n almost up to 4g, finding agreement with the Random Matrix results except for small n and for n=2g. As an application we compute the one-level density of zeros of the zeta function of the curves, including lower-order terms, for test functions whose Fourier transform is supported in (-2,2). The results confirm in part a conjecture of Katz and Sarnak, that to leading order the low-lying zeros for this ensemble have symplectic statistics.

Fibered symplectic cohomology and the Leray-Serre spectral sequence

Abstract: We define Symplectic cohomology groups for a class of symplectic fibrations with closed symplectic base and convex at infinity fiber. The crucial geometric assumption on the fibration is a negativity property reminiscent of negative curvature in complex vector bundles. When the base is symplectically aspherical we construct a spectral sequence of Leray-Serre type converging to the Symplectic cohomology groups of the total space, and we use it to prove new cases of the Weinstein conjecture.