Showing papers on "Symplectic representation published in 2015"

Explicit high-order non-canonical symplectic particle-in-cell algorithms for Vlasov-Maxwell systems

Abstract: Explicit high-order non-canonical symplectic particle-in-cell algorithms for classical particle-field systems governed by the Vlasov-Maxwell equations are developed. The algorithms conserve a discrete non-canonical symplectic structure derived from the Lagrangian of the particle-field system, which is naturally discrete in particles. The electromagnetic field is spatially discretized using the method of discrete exterior calculus with high-order interpolating differential forms for a cubic grid. The resulting time-domain Lagrangian assumes a non-canonical symplectic structure. It is also gauge invariant and conserves charge. The system is then solved using a structure-preserving splitting method discovered by He et al. [preprint arXiv:1505.06076 (2015)], which produces five exactly soluble sub-systems, and high-order structure-preserving algorithms follow by combinations. The explicit, high-order, and conservative nature of the algorithms is especially suitable for long-term simulations of particle-field systems with extremely large number of degrees of freedom on massively parallel supercomputers. The algorithms have been tested and verified by the two physics problems, i.e., the nonlinear Landau damping and the electron Bernstein wave.

Non-Compact Symplectic Toric Manifolds

Abstract: A key result in equivariant symplectic geometry is Delzant's classification of compact connected symplectic toric manifolds. The moment map induces an embedding of the quotient of the manifold by the torus action into the dual of the Lie algebra of the torus; its image is a unimodular ("Delzant") polytope; this gives a bijection between unimodular polytopes and isomorphism classes of compact connected symplectic toric manifolds. In this paper we extend Delzant's classification to non-compact symplectic toric manifolds. For a non-compact symplectic toric manifold the image of the moment map need not be convex and the induced map on the quotient need not be an embedding. Moreover, even when the map on the quotient is an embedding, its image no longer determines the symplectic toric manifold; a degree two characteristic class on the quotient makes an appearance. Nevertheless, the quotient is a manifold with corners, and the induced map from the quotient to the dual of the Lie algebra is what we call a unimodular local embedding. We classify non-compact symplectic toric manifolds in terms of manifolds with corners equipped with degree two cohomology classes and unimodular local embeddings into the dual of the Lie algebra of the corresponding torus. The main new ingredient is the construction of a symplectic toric manifold from such data. The proof passes through an equivalence of categories between symplectic toric manifolds and symplectic toric bundles over a fixed unimodular local embedding. This equivalence also gives a geometric interpretation of the degree two cohomology class.

Wiggling throat of extremal black holes

Abstract: We construct the classical phase space of geometries in the near-horizon region of vacuum extremal black holes as announced in [arXiv:1503.07861]. Motivated by the uniqueness theorems for such solutions and for perturbations around them, we build a family of metrics depending upon a single periodic function defined on the torus spanned by the U(1) isometry directions. We show that this set of metrics is equipped with a consistent symplectic structure and hence defines a phase space. The phase space forms a representation of an infinite dimensional algebra of so-called symplectic symmetries. The symmetry algebra is an extension of the Virasoro algebra whose central extension is the black hole entropy. We motivate the choice of diffeomorphisms leading to the phase space and explicitly derive the symplectic structure, the algebra of symplectic symmetries and the corresponding conserved charges. We also discuss a formulation of these charges with a Liouville type stress-tensor on the torus defined by the U(1) isometries and outline possible future directions.

On symplectic eigenvalues of positive definite matrices

Abstract: If A is a 2n × 2n real positive definite matrix, then there exists a symplectic matrix M such that MTAM=DOOD where D = diag(d1(A), …, dn(A)) is a diagonal matrix with positive diagonal entries, which are called the symplectic eigenvalues of A. In this paper, we derive several fundamental inequalities about these numbers. Among them are relations between the symplectic eigenvalues of A and those of At, between the symplectic eigenvalues of m matrices A1, …, Am and of their Riemannian mean, a perturbation theorem, some variational principles, and some inequalities between the symplectic and ordinary eigenvalues.

Khovanov homology from Floer cohomology

Abstract: This paper realises the Khovanov homology of a link in the 3-sphere as a Lagrangian Floer cohomology group, establishing a conjecture of Seidel and the second author. The starting point is the previously established formality theorem for the symplectic arc algebra over a field k of characteristic zero. Here we prove the symplectic cup and cap bimodules which relate different symplectic arc algebras are themselves formal over k, and construct a long exact triangle for symplectic Khovanov cohomology. We then prove the symplectic and combinatorial arc algebras are isomorphic over the integers in a manner compatible with the cup bimodules. It follows that Khovanov homology and symplectic Khovanov cohomology co-incide in characteristic zero.

Making cobordisms symplectic

Abstract: We establish an existence $h$-principle for symplectic cobordisms of dimension $2n>4$ with concave overtwisted contact boundary

On Thermodynamics and Phase Space of Near Horizon Extremal Geometries

Abstract: Near Horizon Extremal Geometries (NHEG), are geometries which may appear in the near horizon region of the extremal black holes. These geometries have $SL(2,\mathbb{R})\!\times\!U(1)^n$ isometry, and constitute a family of solutions to the theory under consideration. In the first part of this report, their thermodynamic properties are reviewed, and their three universal laws are derived. In addition, at the end of the first part, the role of these laws in black hole thermodynamics is presented. In the second part of this thesis, we review building their classical phase space in the Einstein-Hilbert theory. The elements in the NHEG phase space manifold are built by appropriately chosen coordinate transformations of the original metric. These coordinate transformations are generated by some vector fields, dubbed "symplectic symmetry generators." To fully specify the phase space, we also need to identify the symplectic structure. In order to fix the symplectic structure, we use the formulation of Covariant Phase Space method. The symplectic structure has two parts, the Lee-Wald term and a boundary contribution. The latter is fixed requiring on-shell vanishing of the symplectic current, which guarantees the conservation and integrability of the symplectic structure, and leads to the new concept of "symplectic symmetry." Given the symplectic structure, we construct the corresponding conserved charges, the "symplectic symmetry generators." We also specify the explicit expression of the charges as a functional over the phase space. These symmetry generators constitute the "NHEG algebra," which is an infinite dimensional algebra (may be viewed as a generalized Virasoro), and admits a central extension which is equal to the black hole entropy.

High order symplectic integrators based on continuous-stage Runge-Kutta Nystrom methods

Abstract: On the basis of the previous work by Tang \& Zhang (Appl. Math. Comput. 323, 2018, p. 204--219), in this paper we present a more effective way to construct high-order symplectic integrators for solving second order Hamiltonian equations. Instead of analyzing order conditions step by step as shown in the previous work, the new technique of this paper is using Legendre expansions to deal with the simplifying assumptions for order conditions. With the new technique, high-order symplectic integrators can be conveniently devised by truncating an orthogonal series.

Symplectic homology and the Eilenberg-Steenrod axioms

Abstract: We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and show that it satisfies analogues of the Eilenberg-Steenrod axioms except for the dimension axiom. The resulting long exact sequence of a pair generalizes various earlier long exact sequences such as the handle attaching sequence, the Legendrian duality sequence, and the exact sequence relating symplectic homology and Rabinowitz Floer homology. New consequences of this framework include a Mayer-Vietoris exact sequence for symplectic homology, invariance of Rabinowitz Floer homology under subcritical handle attachment, and a new product on Rabinowitz Floer homology unifying the pair-of-pants product on symplectic homology with a secondary coproduct on positive symplectic homology. In the appendix, joint with Peter Albers, we discuss obstructions to the existence of certain Liouville cobordisms.

On base manifolds of Lagrangian fibrations

Abstract: We consider base spaces of Lagrangian fibrations from singular symplectic varieties. After defining cohomologically irreducible symplectic varieties, we construct an example of Lagrangian fibration whose base space is isomorphic to a quotient of the projective space. We also prove that the base space of Lagrangian fibration from a cohomologically symplectic variety is isomorphic to the projective space provided that the base space is smooth.

Relational Symplectic Groupoids

Abstract: This note introduces the construction of relational symplectic groupoids as a way to integrate every Poisson manifold. Examples are provided and the equivalence, in the integrable case, with the usual notion of symplectic groupoid is discussed.

On locally conformal symplectic manifolds of the first kind

Abstract: We present some examples of locally conformal symplectic structures of the first kind on compact nilmanifolds which do not admit Vaisman metrics. One of these examples does not admit locally conformal Kahler metrics and all the structures come from left-invariant locally conformal symplectic structures on the corresponding nilpotent Lie groups. Under certain topological restrictions related with the compactness of the canonical foliation, we prove a structure theorem for locally conformal symplectic manifolds of the first kind. In the non compact case, we show that they are the product of a real line with a compact contact manifold and, in the compact case, we obtain that they are mapping tori of compact contact manifolds by strict contactomorphisms. Motivated by the aforementioned examples, we also study left-invariant locally conformal symplectic structures on Lie groups. In particular, we obtain a complete description of these structures (with non-zero Lee $1$-form) on connected simply connected nilpotent Lie groups in terms of locally conformal symplectic extensions and symplectic double extensions of symplectic nilpotent Lie groups. In order to obtain this description, we study locally conformal symplectic structures of the first kind on Lie algebras.

Shell-model representations of the proton-neutron symplectic model

Abstract: The representation theory of the recently introduced proton-neutron symplectic model in the many-particle Hilbert space is considered The relation of the Sp(12, R) irreducible representations (irreps) with the shell-model classification of the basis states is considered by extending of the state space to the direct product space of SU p (3) ⊗ SU n (3) irreps, generalizing in this way the Elliott’s SU(3) model for the case of two-component system The Sp(12, R) model appears then as a natural multi-major-shell extension of the generalized proton-neutron SU(3) scheme, which takes into account the core collective excitations of monopole and quadrupole, as well as dipole type associated with the giant resonance vibrational degrees of freedom Each Sp(12, R) irreducible representation is determined by a symplectic bandhead or an intrinsic U(6) space which can be fixed by the underlying proton-neutron shell-model structure, so the theory becomes completely compatible with the Pauli principle It is shown that this intrinsic U(6) structure is of vital importance for the appearance of the low-lying collective bands without involving a mixing of different symplectic irreps The full range of low-lying collective states can then be described by the microscopically based intrinsic U(6) structure, renormalized by coupling to the giant resonance vibrations

Symplectic dilations, Gaussian states and Gaussian channels

Abstract: By elementary matrix algebra we show that every real 2n×2n matrix admits a dilation to an element of the real symplectic group Sp(2(n+m)) for some nonnegative integer m. Our methods do not yield the minimum value of m, for which such a dilation is possible.

Codimension-one Symplectic Foliations : Constructions and Examples

Abstract: This thesis addresses the question: which compact manifolds admit codimension-one symplectic foliations? It develops a method to construct such symplectic foliations on compact manifolds, called the “turbulisation method”. This method is applied then for the construction of such symplectic foliations on manifolds admitting certain type of open book decompositions and on products of the circle with manifolds admitting achiral Lefschetz fibrations. This applications allow us to enlarge the class of manifolds for which the answer to the main question is positive. The method relies on the existence of certain symplectic structures that are “constant” around the boundary. These symplectic structures are related with a special type of Poisson structures, called the log-symplectic structures. In the last part of the thesis, we study and characterise the space of deformations of log-symplectic structures.

Extremal rays and automorphisms of holomorphic symplectic varieties

Abstract: We survey recent results on ample cones and birational contractions of holomorphic symplectic varieties of K3 type, focusing on explicit constructions and concrete examples.

Symmetries of order four on K3 surfaces

Abstract: We study automorphisms of order four on K3 surfaces. The symplectic ones have been first studied by Nikulin, they are known to fix six points and their action on the K3 lattice is unique. In this paper we give a classification of the purely non-symplectic automorphisms by relating the structure of their fixed locus to their action on cohomology, in the following cases: the fixed locus contains a curve of genus g>0; the fixed locus contains at least a curve and all the curves fixed by the square of the automorphism are rational. We give partial results in the other cases. Finally, we classify non-symplectic automorphisms of order four with symplectic square.

Elliptic singularities on log symplectic manifolds and Feigin--Odesskii Poisson brackets

Abstract: A log symplectic manifold is a complex manifold equipped with a complex symplectic form that has simple poles on a hypersurface The possible singularities of such a hypersurface are heavily constrained We introduce the notion of an elliptic point of a log symplectic structure, which is a singular point at which a natural transversality condition involving the modular vector field is satisfied, and we prove a local normal form for such points that involves the simple elliptic surface singularities $\tilde{E}_6,\tilde{E}_7$ and $\tilde{E}_8$ Our main application is to the classification of Poisson brackets on Fano fourfolds For example, we show that Feigin and Odesskii's Poisson structures of type $q_{5,1}$ are the only log symplectic structures on projective four-space whose singular points are all elliptic

Cox rings of some symplectic resolutions of quotient singularities

Abstract: We investigate Cox rings of symplectic resolutions of quotients of $\mathbb{C}^{2n}$ by finite symplectic group actions. We propose a finite generating set of the Cox ring of a symplectic resolution and prove that under a condition concerning monomial valuations it is sufficient. Also, three 4-dimensional examples are described in detail. Generators of the (expected) Cox rings of symplectic resolutions are computed and in one case a resolution is constructed as a GIT quotient of the spectrum of the Cox ring.

Symplectic realizations of holomorphic Poisson manifolds

Abstract: Symplectic realization is a longstanding problem which can be traced back to Sophus Lie. In this paper, we present an explicit solution to this problem for an arbitrary holomorphic Poisson manifold. More precisely, for any holomorphic Poisson manifold $(X, \pi)$, we prove that there exists a holomorphic symplectic structure in a neighborhood $Y$ of the zero section of $T^*X$ such that the projection map is a symplectic realization of the given Poisson manifold, and moreover the zero section is a holomorphic Lagrangian submanifold. We describe an explicit construction for such a new holomorphic symplectic structure on $Y \subseteq T^*X$.

A new solution of Schrödinger equation based on symplectic algorithm

Abstract: An explicit three dimensional symplectic finite-difference time-domain method is introduced to solve the Schrodinger equation. The method is obtained by using a symplectic scheme with a propagator of exponential differential operators for the time direction approximation and fourth-order collocated finite differences for the space discretization. Firstly, a high-order symplectic framework for discretizing the Schrodinger equation is presented. Secondly, the comparisons on numerical stability, dispersion are also provided between the new symplectic scheme and other commonly used methods. Finally, numerical examples are given to evaluate the performance of the proposed method and the advantages on the accuracy and stability are also further demonstrated.

Higher hairy graph homology

Abstract: We study the hairy graph homology of a cyclic operad; in particular we show how to assemble corresponding hairy graph cohomology classes to form cocycles for ordinary graph homology, as defined by Kontsevich. We identify the part of hairy graph homology coming from graphs with cyclic fundamental group as the dihedral homology of a related associative algebra with involution. For the operads $$\mathsf{Comm}$$ $$\mathsf{Assoc}$$ and $$\mathsf{Lie}$$ we compute this algebra explicitly, enabling us to apply known results on dihedral homology to the computation of hairy graph homology. In addition we determine the image in hairy graph homology of the trace map defined in Conant et al. (J Topology 6(1):119–153, 2013), as a symplectic representation. For the operad $$\mathsf{Lie}$$ assembling hairy graph cohomology classes yields all known non-trivial rational homology of $$Out(F_n)$$ . The hairy graph homology of $$\mathsf{Lie}$$ is also useful for constructing elements of the cokernel of the Johnson homomorphism of a once-punctured surface.

Contraction of Hamiltonian $K$-spaces

Abstract: In the spirit of recent work of Harada-Kaveh and Nishinou-Nohara-Ueda, we study the symplectic geometry of Popov's horospherical degenerations of complex algebraic varieties with the action of a complex linearly reductive group. We formulate an intrinsic symplectic contraction of a Hamiltonian space, which is a surjective, continuous map onto a new Hamiltonian space that is a symplectomorphism on an explicitly defined dense open subspace. This map is given by a precise formula, using techniques from the theory of symplectic reduction and symplectic implosion. We then show, using the Vinberg monoid, that the gradient-Hamiltonian flow for a horospherical degeneration of an algebraic variety gives rise to this contraction from a general fiber to the special fiber. We apply this construction to branching problems in representation theory, and finally we show how the Gel'fand-Tsetlin integrable system can be understood to arise this way.

Bernstein inequality and holonomic modules

Abstract: In this paper we study the representation theory of filtered algebras with commutative associated graded whose spectrum has finitely many symplectic leaves. Examples are provided by the algebras of global sections of quantizations of symplectic resolutions, quantum Hamiltonian reductions, spherical symplectic reflection algebras. We introduce the notion of holonomic modules for such algebras. We show that the generalized Bernstein inequality holds for simple modules and turns into equality for holonomic simples provided the algebraic fundamental groups of all leaves are finite. Under the same assumption, we prove that the associated variety of a simple holonomic module is equi-dimensional. We also prove that, if the regular bimodule has finite length or if the algebra in question is a quantum Hamiltonian reduction, then any holonomic module has finite length. This allows to reduce the Bernstein inequality for arbitrary modules to simple ones. We prove that the regular bimodule has finite length for the global sections of quantizations of symplectic resolutions and for Rational Cherednik algebras. The paper contains a joint appendix by the author and Etingof that motivates the definition of a holonomic module in the case of global sections of a quantization of a symplectic resolution.