Showing papers on "Symplectic manifold published in 2015"
TL;DR: In this article, an explicit high-order non-canonical symplectic particle-in-cell algorithm for classical particle-field systems governed by the Vlasov-Maxwell equations is developed.
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.
88 citations
TL;DR: In this paper, the authors extend Delzant's classification to non-compact symplectic toric manifolds and construct a manifold with corners equipped with degree two cohomology classes and unimodular local embeddings into the dual of the corresponding torus.
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.
70 citations
TL;DR: In this paper, a family of metrics based on a single periodic function defined on the torus spanned by the U(1) isometry directions was constructed and a phase space of geometries in the nearhorizon region of vacuum extremal black holes was defined.
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.
55 citations
TL;DR: In this paper, the authors derived several fundamental inequalities about the eigenvalues of 2n × 2n real positive definite matrices, such as the relation between the symplectic eigenvalue of A and those of At, between the Riemannian mean of m matrices A1, A2, Am, and Am and their mean of their mean, a perturbation theorem, and some variational principles.
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.
45 citations
TL;DR: The notion of normal crossing divisors from algebraic geometry has been introduced in this paper, where the authors define the GW invariant of a symplectic manifold X relative to such a divisor V. The invariants are key ingredients in symplectic sum type formulas for GW invariants, and extend those defined in our previous joint work with T.H. Parker [16].
43 citations
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TL;DR: In this paper, it was shown that the Khovanov homology of a link in the 3-sphere as a Lagrangian Floer cohomology group is a conjecture of Seidel and the second author.
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.
42 citations
TL;DR: In this article, a quantization commutes with reduction principle was formulated for non-compact Lie groups and manifolds, where the action is proper and the zero set of a deformation vector field, associated to the momentum map and an equivariant family of inner products on the Lie algebra g of G, is G-cocompact.
41 citations
TL;DR: This criterion is applied to the classification of non-symplectic prime order automorphisms on manifolds of K3-type and it is proved that almost all cases are covered.
Abstract: We introduce the notion of induced automorphisms in order to state a criterion to determine whether a given automorphism on a manifold of $K3^{[n]}$ type is, in fact, induced by an automorphism of a $K3$ surface and the manifold is a moduli space of stable objects on the $K3$. This criterion is applied to the classification of non-symplectic prime order automorphisms on manifolds of $K3^{[2]}$ type and we prove that almost all cases are covered. Variations of this notion and the above criterion are introduced and discussed for the other known deformation types of irreducible symplectic manifolds. Furthermore we provide a description of the Picard lattice of several irreducible symplectic manifolds having a lagrangian fibration.
40 citations
TL;DR: In this article, the authors constructed a compact simply-connected 7-dimensional manifold admitting a K-contact structure but not a Sasakian structure, and proved that such a manifold has vanishing cup-product on the second cohomology.
Abstract: We construct a compact simply-connected 7-dimensional manifold admitting a K-contact structure but not a Sasakian structure. We also study rational homotopy properties of such manifolds, proving in particular that a simply-connected 7-dimensional Sasakian manifold has vanishing cup-product on the second cohomology and that it is formal if and only if all its triple Massey products vanish.
31 citations
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TL;DR: In this article, the Lagrangian submanifolds of a monotone Lagrangians of a symplectic manifold were constructed using canonical orientations, with no further choices such as relative Pin-structures.
Abstract: The purpose of this paper is to extend the construction of the PSS-type isomorphism between the Floer homology and the quantum homology of a monotone Lagrangian submanifold $L$ of a symplectic manifold $M$, provided that the minimal Maslov number of $L$ is at least two, to arbitrary coefficients. We provide a proof, again over arbitrary coefficients, that this isomorphism respects the natural algebraic structures on both sides, such as the quantum product and the quantum module action. This isomorphism serves as the technical foundation for the construction of Lagrangian spectral invariants in a joint paper with Remi Leclercq (arXiv:1505:07430). Our constructions work when the second Stiefel--Whitney class of $L$ vanishes on the image of the boundary homomorphism $\pi_3(M,L) \to \pi_2(L)$, a condition strictly weaker than being relatively Pin; in particular we do not require $L$ to be orientable. The constructions are done using canonical orientations, and require no further choices such as relative Pin-structures. Such structures do however play a significant role when endowing the various complexes and homologies with structures of modules over Novikov rings, and in calculations.
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TL;DR: In this article, the authors considered the near horizon extremal geometries (NHEG) and derived three universal laws for their thermodynamic properties and the role of these laws in black hole thermodynamics.
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.
TL;DR: In this paper, a regularized index of a generalized Dirac operator on a complete Riemannian manifold endowed with a proper action of a unimodular Lie group was constructed.
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TL;DR: Tang et al. as mentioned in this paper presented a more effective way to construct high-order symplectic integrators for solving second order Hamiltonian equations by using Legendre expansions to deal with the simplifying assumptions for order conditions.
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.
TL;DR: In this paper, the authors 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.
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.
TL;DR: In this paper, the authors considered the base space of Lagrangian fibrations from singular symplectic varieties and proved that the Lagrangians' base space is a quotient of the projective space.
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.
01 Jan 2015
TL;DR: In this article, the authors define the perverse sheaf of vanishing cycles (PV{U,f}$, a perverse sheaf on a regular function, and show that it depends up to canonical isomorphism only on the critical locus of the function.
Abstract: Let $U$ be a smooth $\mathbb C$-scheme, $f:U\to\mathbb A^1$ a regular function, and $X=$Crit$(f)$ the critical locus, as a $\mathbb C$-subscheme of $U$. Then one can define the "perverse sheaf of vanishing cycles" $PV_{U,f}$, a perverse sheaf on $X$. This paper proves four main results: (a) Suppose $\Phi:U\to U$ is an isomorphism with $f\circ\Phi=f$ and $\Phi\vert_X=$id$_X$. Then $\Phi$ induces an isomorphism $\Phi_*:PV_{U,f}\to PV_{U,f}$. We show that $\Phi_*$ is multiplication by det$(d\Phi\vert_X)=1$ or $-1$. (b) $PV_{U,f}$ depends up to canonical isomorphism only on $X^{(3)},f^{(3)}$, for $X^{(3)}$ the third-order thickening of $X$ in $U$, and $f^{(3)}=f\vert_{X^{(3)}}:X^{(3)}\to\mathbb A^1$. (c) If $U,V$ are smooth $\mathbb C$-schemes, $f:U\to\mathbb A^1$, $g:V\to\mathbb A^1$ are regular, $X=$Crit$(f)$, $Y=$Crit$(g)$, and $\Phi:U\to V$ is an embedding with $f=g\circ\Phi$ and $\Phi\vert_X:X\to Y$ an isomorphism, there is a natural isomorphism $\Theta_\Phi:PV_{U,f}\to\Phi\vert_X^*(PV_{V,g})\otimes_{\mathbb Z_2}P_\Phi$, for $P_\Phi$ a natural principal $\mathbb Z_2$-bundle on $X$. (d) If $(X,s)$ is an oriented d-critical locus in the sense of Joyce arXiv:1304.4508, there is a natural perverse sheaf $P_{X,s}$ on $X$, such that if $(X,s)$ is locally modelled on Crit$(f:U\to\mathbb A^1)$ then $P_{X,s}$ is locally modelled on $PV_{U,f}$. We also generalize our results to replace $U,X$ by $\mathbb K$-schemes over other fields $\mathbb K$, or by complex analytic spaces, and $PV_{U,f}$ by $\mathcal D$-modules, or mixed Hodge modules. We discuss applications of (d) to categorifying Donaldson-Thomas invariants of Calabi-Yau 3-folds, and to defining a 'Fukaya category' of Lagrangians in a complex symplectic manifold using perverse sheaves. This is the third in a series of papers arXiv:1304.4508, arXiv:1305.6302, arXiv:1305.6428, arXiv:1312.0090.
TL;DR: In this article, the authors introduce relational symplectic groupoids as a way to integrate every Poisson manifold, and the equivalence of relational groupoids with the usual notion of symplectic groupsoid is discussed.
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.
TL;DR: In this article, the authors studied locally conformal symplectic structures of the first kind on compact nilmanifolds which do not admit Vaisman metrics and obtained a complete description of these structures (with non-zero Lee $1$-form) on connected simply connected nilpotent Lie groups.
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.
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TL;DR: In this article, it was shown that blowups or reverse flips of rational symplectic manifolds with point centers create Floer-non-trivial Lagrangian tori.
Abstract: We show that blow-ups or reverse flips (in the sense of the minimal model program) of rational symplectic manifolds with point centers create Floer-non-trivial Lagrangian tori. As applications, we demonstrate the existence of Hamiltonian non-displaceable Lagrangian tori in, for example, small symplectic blow-ups of compact symplectic manifolds and moduli spaces of polygons. These results are part of a conjectural description of generators for the Fukaya category of a compact symplectic manifold with a singularity-free running of the minimal model program.
Book•
25 Sep 2015
TL;DR: In this paper, Griffiths' theory of period mappings for compact Kahler manifolds is extended and applied to families of open complex manifolds, and the degeneration behavior of the relative Frolicher spectral sequence associated to a submersive morphism of complex manifold is investigated.
Abstract: Extending Griffiths' classical theory of period mappings for compact Kahler manifolds, this book develops and applies a theory of period mappings of "Hodge-de Rham type" for families of open complex manifolds. The text consists of three parts. The first part develops the theory. The second part investigates the degeneration behavior of the relative Frolicher spectral sequence associated to a submersive morphism of complex manifolds. The third part applies the preceding material to the study of irreducible symplectic complex spaces. The latter notion generalizes the idea of an irreducible symplectic manifold, dubbed an irreducible hyperkahler manifold in differential geometry, to possibly singular spaces. The three parts of the work are of independent interest, but intertwine nicely.
TL;DR: In this paper, the Maslov (P, ω)-index theory for a symplectic path is developed and the Bott-type iteration formula is proved, and it is shown that the Bott formula can be used for a manifold path.
Abstract: Abstract In this paper, the Maslov (P, ω)-index theory for a symplectic path is developed and the Bott-type iteration formula is proved.
TL;DR: Gustafson, Halmos, and Radjavi as discussed by the authors proved that every nonsingular even size matrix is a product of skew symmetric matrices, and they proved that these coninvolutions can be taken to be symplectic if A is symplectic (every symplectic matrix has unit determinant).
TL;DR: In this article, it was shown 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.
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.
TL;DR: In this article, the authors studied holomorphic disc with boundary on a Lagrangian submanifold and proved that it has non-vanishing Floer cohomology if and only if the coefficient ring has characteristic 5.
Abstract: We study holomorphic discs with boundary on a Lagrangian submanifold $$L$$
in a symplectic manifold admitting a Hamiltonian action of a group $$K$$
which has $$L$$
as an orbit. We prove various transversality and classification results for such discs which we then apply to the case of a particular Lagrangian in $$\mathbf {C}\mathbf {P}^3$$
first noticed by Chiang (Int Math Res Not 45:2437–2441, 2004). We prove that this Lagrangian has non-vanishing Floer cohomology if and only if the coefficient ring has characteristic 5, in which case, it generates the split-closed derived Fukaya category as a triangulated category.
TL;DR: In this article, a topological sigma model and a current algebra based on a Courant algebroid structure on a Poisson manifold were constructed by supergeometric construction on a QP manifold.
Abstract: We construct a topological sigma model and a current algebra based on a Courant algebroid structure on a Poisson manifold. In order to construct models, we reformulate the Poisson Courant algebroid by supergeometric construction on a QP-manifold. A new duality of Courant algebroids which transforms H-flux and R-flux is proposed, where the transformation is interpreted as a canonical transformation of a graded symplectic manifold.
30 Sep 2015
Dissertation•
28 Sep 2015
TL;DR: In this article, the Turbulization method is applied for the construction of such symplectic foliations on manifolds admitting certain type of open book decompositions and on products of the circle with manifold admitting achiral Lefschetz fibrations, which allows to enlarge the class of manifolds for which the answer to the main question is positive.
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.
TL;DR: In this paper, it was shown that for most unitary K-modules the corresponding symplectic quotient is not regularly symplectomorphic to a linear symplectic orbifold (the quotient of a unitary module of a finite group).
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TL;DR: In this article, the authors define a class of manifolds called universal symplectic cohomology, which includes all geometrically bounded symplectic manifolds and develop a formalism for Floer theory.
Abstract: We define a class of symplectic manifolds which includes all geometrically bounded symplectic manifolds. In this class we develop a formalism for Floer theory. We show that such a manifold carries a cofinal symplectically invariant set of Floer data which allow the definition of Floer complexes. Moreover, the set of all such Floer complexes forms a directed system. This gives rise to a symplectic invariant we call universal symplectic cohomology. We show that symplectic cohomology as hitherto defined for Liouville domains coincides with universal symplectic cohomology. We discuss applications to the problem of nearby existence of periodic orbits. The results rest on a novel approach to controlling the diameters of Floer trajectories.