Showing papers on "Symplectic representation published in 2016"

Symplectic and Killing symmetries of AdS3 gravity: holographic vs boundary gravitons

Abstract: The set of solutions to the AdS3 Einstein gravity with Brown-Henneaux boundary conditions is known to be a family of metrics labeled by two arbitrary periodic functions, respectively left and right-moving. It turns out that there exists an appropriate presymplectic form which vanishes on-shell. This promotes this set of metrics to a phase space in which the Brown-Henneaux asymptotic symmetries become symplectic symmetries in the bulk of spacetime. Moreover, any element in the phase space admits two global Killing vectors. We show that the conserved charges associated with these Killing vectors commute with the Virasoro symplectic symmetry algebra, extending the Virasoro symmetry algebra with two U(1) generators. We discuss that any element in the phase space falls into the coadjoint orbits of the Virasoro algebras and that each orbit is labeled by the U(1) Killing charges. Upon setting the right-moving function to zero and restricting the choice of orbits, one can take a near-horizon decoupling limit which preserves a chiral half of the symplectic symmetries. Here we show two distinct but equivalent ways in which the chiral Virasoro symplectic symmetries in the near-horizon geometry can be obtained as a limit of the bulk symplectic symmetries.

Symplectic Model Reduction of Hamiltonian Systems

Abstract: In this paper, a symplectic model reduction technique, proper symplectic decomposition (PSD) with symplectic Galerkin projection, is proposed to save the computational cost for the simplification of large-scale Hamiltonian systems while preserving the symplectic structure. As an analogy to the classical proper orthogonal decomposition (POD)-Galerkin approach, PSD is designed to build a symplectic subspace to fit empirical data, while the symplectic Galerkin projection constructs a reduced Hamiltonian system on the symplectic subspace. For practical use, we introduce three algorithms for PSD, which are based upon the cotangent lift, complex singular value decomposition, and nonlinear programming. The proposed technique has been proven to preserve system energy and stability. Moreover, PSD can be combined with the discrete empirical interpolation method to reduce the computational cost for nonlinear Hamiltonian systems. Owing to these properties, the proposed technique is better suited than the classical PO...

Explicit symplectic algorithms based on generating functions for charged particle dynamics

Abstract: Dynamics of a charged particle in the canonical coordinates is a Hamiltonian system, and the well-known symplectic algorithm has been regarded as the de facto method for numerical integration of Hamiltonian systems due to its long-term accuracy and fidelity. For long-term simulations with high efficiency, explicit symplectic algorithms are desirable. However, it is generally believed that explicit symplectic algorithms are only available for sum-separable Hamiltonians, and this restriction limits the application of explicit symplectic algorithms to charged particle dynamics. To overcome this difficulty, we combine the familiar sum-split method and a generating function method to construct second- and third-order explicit symplectic algorithms for dynamics of charged particle. The generating function method is designed to generate explicit symplectic algorithms for product-separable Hamiltonian with form of H(x,p)=p_{i}f(x) or H(x,p)=x_{i}g(p). Applied to the simulations of charged particle dynamics, the explicit symplectic algorithms based on generating functions demonstrate superiorities in conservation and efficiency.

Quantization of Compact Symplectic Manifolds

Abstract: We develop the theory of Berezin–Toeplitz operators on any compact symplectic prequantizable manifold from scratch. Our main inspiration is the Boutet de Monvel–Guillemin theory, which we simplify in several ways to obtain a concise exposition. A comparison with the spin-c Dirac quantization is also included.

Deformations of symplectic singularities and Orbit method for semisimple Lie algebras

Abstract: We classify filtered quantizations of conical symplectic singularities and use this to show that all filtered quantizations of symplectic quotient singularities are spherical Symplectic reflection algebras of Etingof and Ginzburg. We further apply our classification and a classification of filtered Poisson deformations obtained by Namikawa to establish a version of the Orbit method for semisimple Lie algebras. Namely, we produce a natural map from the set of adjoint orbits in a semisimple Lie algebra to the set of primitive ideals in the universal enveloping algebra. We show that the map is injective for classical Lie algebras.

A global Torelli theorem for singular symplectic varieties

Abstract: We systematically study the moduli theory of symplectic varieties (in the sense of Beauville) which admit a resolution by an irreducible symplectic manifold. In particular, we prove an analog of Verbitsky's global Torelli theorem for the locally trivial deformations of such varieties. Verbitsky's work on ergodic complex structures replaces twistor lines as the essential global input. In so doing we extend many of the local deformation-theoretic results known in the smooth case to such (not-necessarily-projective) symplectic varieties. We deduce a number of applications to the birational geometry of symplectic manifolds, including some results on the classification of birational contractions of $K3^{[n]}$-type varieties.

High-Order Symplectic Partitioned Lie Group Methods

Abstract: In this article, a unified approach to obtain symplectic integrators on $$T^{*}G$$T?G from Lie group integrators on a Lie group $$G$$G is presented. The approach is worked out in detail for symplectic integrators based on Runge---Kutta---Munthe-Kaas methods and Crouch---Grossman methods. These methods can be interpreted as symplectic partitioned Runge---Kutta methods extended to the Lie group setting in two different ways. In both cases, we show that it is possible to obtain symplectic integrators of arbitrarily high order by this approach.

On the (non)existence of symplectic resolutions of linear quotients

Abstract: We study the existence of symplectic resolutions of quotient singularities V/GV/G, where VV is a symplectic vector space and GG acts symplectically. Namely, we classify the symplectically irreducible and imprimitive groups, excluding those of the form K⋊S2K⋊S2 where K

A minimal-variable symplectic integrator on spheres

Abstract: We construct a symplectic, globally defined, minimal-variable, equivariant integrator on products of 2-spheres. Examples of corresponding Hamiltonian systems, called spin systems, include the reduced free rigid body, the motion of point vortices on a sphere, and the classical Heisenberg spin chain, a spatial discretisation of the Landau-Lifshitz equation. The existence of such an integrator is remarkable, as the sphere is neither a vector space, nor a cotangent bundle, has no global coordinate chart, and its symplectic form is not even exact. Moreover, the formulation of the integrator is very simple, and resembles the geodesic midpoint method, although the latter is not symplectic.

A characterization of nilpotent orbit closures among symplectic singularities II

Abstract: This short note is a supplement to the previous article with the same title. Here we treat a conical symplectic variety obtained as a finite covering of a (not necessarily normal) nilpotent orbit closure of a complex semisimple Lie algebra.

Shifted Symplectic and Poisson Structures on Spaces of Framed Maps

Abstract: We examine shifted symplectic and Poisson structures on spaces of framed maps. We prove some results about shifted Poisson structures analogous to those in existing ones about symplectic structures. Then, we consider the space Map(X,D,Y) of maps from X to Y framed along a divisor D. We give conditions under which this space has a shifted symplectic or Poisson structure. Classical examples of symplectic and Poisson structures are provided with this theorem.

A recursive basis for primitive forms in symplectic spaces and applications to Heisenberg groups

Abstract: This paper is divided in two parts: in Section 2, we define recursively a privileged basis of the primitive forms in a symplectic space (V2n, ω). Successively, in Section 3, we apply our construction in the setting of Heisenberg groups ℍn, n ≥ 1, to write in coordinates the exterior differential of the so-called Rumin’s complex of differential forms in ℍn.

Reduction of symplectic homeomorphisms

Abstract: In our previous article, we proved that symplectic homeomorphisms preserving a coisotropic submanifold C, preserve its characteristic foliation as well. As a consequence, such symplectic homeomorphisms descend to the reduction of the coisotropic C. In this article we show that these reduced homeomorphisms continue to exhibit certain symplectic properties. In particular, in the specific setting where the symplectic manifold is a torus and the coisotropic is a standard subtorus, we prove that the reduced homeomorphism preserves spectral invariants and hence the spectral capacity.

Conformal symplectic geometry of cotangent bundles

Abstract: We prove a version of the Arnol'd conjecture for Lagrangian submanifolds of conformal symplectic manifolds: a Lagrangian $L$ which has non-zero Morse-Novikov homology for the restriction of the Lee form $\beta$ cannot be disjoined from itself by a $C^0$-small Hamiltonian isotopy. Furthermore for generic such isotopies the number of intersection points equals at least the sum of the free Betti numbers of the Morse-Novikov homology of $\beta$. We also give a short exposition of conformal symplectic geometry, aimed at readers who are familiar with (standard) symplectic or contact geometry.

A duality map for the quantum symplectic double

Abstract: We consider a cluster variety called the symplectic double, defined for an oriented disk with finitely many marked points on its boundary. We construct a canonical map from the tropical integral points of this cluster variety into its quantized algebra of rational functions. As a special case, we obtain a solution to Fock and Goncharov's duality conjectures for quantum cluster varieties associated to a disk with marked points. This extends the author's previous work with Kim on quantum cluster varieties associated to punctured surfaces.

Symplectic, Poisson, and contact geometry on scattering manifolds

Abstract: We introduce scattering-symplectic manifolds, manifolds with a type of minimally degenerate Poisson structure that is not too restrictive so as to have a large class of examples, yet restrictive enough for standard Poisson invariants to be computable. This paper will demonstrate the potential of the scattering symplectic setting. In particular, we construct scattering-symplectic spheres and scattering symplectic gluings between strong convex symplectic fillings of a contact manifold. By giving an explicit computation of the Poisson cohomology of a scattering symplectic manifold, we also introduce a new method of computing Poisson cohomology.

Manifolds Which Are Complex and Symplectic But Not Kähler

Abstract: The first example of a compact manifold admitting both complex and symplectic structures but not admitting a Kahler structure is the renowned Kodaira–Thurston manifold. We review its construction and show that this paradigm is very general and is not related to the fundamental group. More specifically, we prove that the simply connected eight-dimensional compact manifold of Fernandez and Munoz (Ann Math (2), 167(3):1045–1054, 2008) admits both symplectic and complex structures but does not carry Kahler metrics.

Perversely categorified Lagrangian correspondences

Abstract: In this article, we construct a $2$-category of Lagrangians in a fixed shifted symplectic derived stack S. The objects and morphisms are all given by Lagrangians living on various fiber products. A special case of this gives a $2$-category of $n$-shifted symplectic derived stacks $Symp^n$. This is a $2$-category version of Weinstein's symplectic category in the setting of derived symplectic geometry. We introduce another $2$-category $Symp^{or}$ of $0$-shifted symplectic derived stacks where the objects and morphisms in $Symp^0$ are enhanced with orientation data. Using this, we define a partially linearized $2$-category $LSymp$. Joyce and his collaborators defined a certain perverse sheaf on any oriented $(-1)$-shifted symplectic derived stack. In $LSymp$, the $2$-morphisms in $Symp^{or}$ are replaced by the hypercohomology of the perverse sheaf assigned to the $(-1)$-shifted symplectic derived Lagrangian intersections. To define the compositions in $LSymp$ we use a conjecture by Joyce, that Lagrangians in $(-1)$-shifted symplectic stacks define canonical elements in the hypercohomology of the perverse sheaf over the Lagrangian. We refine and expand his conjecture and use it to construct $LSymp$ and a $2$-functor from $Symp^{or}$ to $LSymp$. We prove Joyce's conjecture in the most general local model. Finally, we define a $2$-category of $d$-oriented derived stacks and fillings. Taking mapping stacks into a $n$-shifted symplectic stack defines a $2$-functor from this category to $Symp^{n-d}$.

The reachable set of single-mode unstable quadratic Hamiltonians

Abstract: The question of open-loop control in the Gaussian regime may be cast by asking which Gaussian unitary transformations are reachable by turning on and off a given set of quadratic Hamiltonians. For compact groups, including finite dimensional unitary groups, the well known Lie algebra rank criterion provides a sufficient and necessary condition for the reachable set to cover the whole group. Because of the non-compact nature of the symplectic group, which corresponds to Gaussian unitary transformations, this criterion turns out to be still necessary but not sufficient for Gaussian systems. If the control Hamiltonians are unstable, in a sense made rigorous in the main text, the peculiar situation may arise where the rank criterion is satisfied and yet not all symplectic transformations are reachable. Here, we address this situation for one degree of freedom and study the properties of the reachable set under unstable control Hamiltonians. First, we provide a partial analytical characterisation of the reachable set and prove that no orthogonal (`energy-preserving' or `passive' in the literature) symplectic operations may be reached with such controls. Then, we apply numerical optimal control algorithms to demonstrate a complete characterisation of the set in specific cases.

On Symplectic Dynamics

Abstract: This paper continues to carry out a foundational study of Banyaga topologies of a closed symplectic manifold [3]. Our intension in writing this paper is to provide several symplectic analogues of some results found in the study of Hamiltonian dynamics. Especially, without appealing to the positivity of the symplectic displacement energy, we point out the impact of the $L^\infty$ version of Banyaga Hofer-like metric in the investigation of the symplectic nature of the $C^0-$limit of a sequence of symplectic maps. This result is the symplectic analogue of a result that was proved in Hofer-Zehnder [8] (for compactly supported Hamiltonian diffeomorphisms on $\mathbb{R}^{2n}$), and then reformulated in Oh-Muller [10] for Hamiltonian diffeomorphisms in general. Furthermore, we extend to symplectic isotopies the regularization procedure for Hamiltonian paths introduced in Polterovich [11], and then we use it to prove the equality between the two versions of Banyaga Hofer-like norms defined on the identity component in the group of symplectomorphisms. This result was announced in [2]. It shows the uniqueness of Banyaga Hofer-like geometry, and then yields the symplectic analogue of a result that was proved in Polterovich [11]. Finally, we elaborate the symplectic analogues of some approximation results found in Oh-Muller [10], and make some remarks on flux theory.

Periodic symplectic cohomologies and obstructions to exact Lagrangian immersions

Abstract: Periodic Symplectic Cohomologies and Obstructions to Exact Lagrangian Immersions Jingyu Zhao Given a Liouville manifold (M, θ), symplectic cohomology is defined as the Hamiltonian Floer homology for the symplectic action functional on the free loop space LM := Maps(S,M). In this thesis, we propose two versions of periodic S-equivariant homology or S-equivariant Tate homology for the natural S-action on the free loop space LM . The first version, denoted as PSH∗(M), is called periodic symplectic cohomology. We prove that there is a localization theorem or a fix point property for PSH∗(M). The second version PSH∗(M) is called the completed periodic symplectic cohomology which satisfies Goodwillie’s excision isomorphism. Inspired by the work of Seidel and Solomon on the existence of dilations on symplectic cohomology, we formulate the notion of Liouville manifolds admitting higher dilations using Goodwillie’s excision isomorphism on the completed periodic symplectic cohomology PSH∗(M). As an application, we derive obstructions to existence of certain exact Lagrangian immersions in Liouville manifolds admitting higher dilations.