Showing papers on "Symplectic representation published in 2001"
Book•
17 Jul 2001TL;DR: Symplectic Toric Manifolds as mentioned in this paper are a type of complex manifold that is composed of two or more complex manifold types, and they can be classified into three classes: symplectic, symplectomorphisms, and symmlectic reduction.
Abstract: Symplectic Manifolds.- Symplectic Forms.- Symplectic Form on the Cotangent Bundle.- Symplectomorphisms.- Lagrangian Submanifolds.- Generating Functions.- Recurrence.- Local Forms.- Preparation for the Local Theory.- Moser Theorems.- Darboux-Moser-Weinstein Theory.- Weinstein Tubular Neighborhood Theorem.- Contact Manifolds.- Contact Forms.- Contact Dynamics.- Compatible Almost Complex Structures.- Almost Complex Structures.- Compatible Triples.- Dolbeault Theory.- Kahler Manifolds.- Complex Manifolds.- Kahler Forms.- Compact Kahler Manifolds.- Hamiltonian Mechanics.- Hamiltonian Vector Fields.- Variational Principles.- Legendre Transform.- Moment Maps.- Actions.- Hamiltonian Actions.- Symplectic Reduction.- The Marsden-Weinstein-Meyer Theorem.- Reduction.- Moment Maps Revisited.- Moment Map in Gauge Theory.- Existence and Uniqueness of Moment Maps.- Convexity.- Symplectic Toric Manifolds.- Classification of Symplectic Toric Manifolds.- Delzant Construction.- Duistermaat-Heckman Theorems.
538 citations
TL;DR: In this paper, it was shown that many important affine quiver varieties can be imbedded as coadjoint orbits in the dual of an appropriate infinite dimensional Lie algebra, where there is an infinitesimally transitive action of the Lie algebra in question on the quiver variety.
Abstract: Quiver varieties have recently appeared in various different areas of Mathematics such as representation theory of Kac-Moody algebras and quantum groups, instantons on 4-manifolds, and resolutions Kleinian singularities. In this paper, we show that many important affine quiver varieties, e.g., the Calogero-Moser space, can be imbedded as coadjoint orbits in the dual of an appropriate infinite dimensional Lie algebra. In particular, there is an infinitesimally transitive action of the Lie algebra in question on the quiver variety. Our construction is based on an extension of Kontsevich's formalism of `non-commutative Symplectic geometry'. We show that this formalism acquires its most adequate and natural formulation in the much more general framework of P-geometry, a `non-commutative geometry' for an algebra over an arbitrary cyclic Koszul operad.
140 citations
TL;DR: In this paper, it was shown that if M is minimal and has b+ = 1, there is a unique canonical class up to sign, and any real second cohomology class of positive square is represented by symplectic forms.
Abstract: Let M be a closed oriented smooth 4-manifold admitting symplectic structures. If M is minimal and has b+ = 1, we prove that there is a unique symplectic canonical class up to sign, and any real second cohomology class of positive square is represented by symplectic forms. Similar results hold when M is not minimal.
121 citations
TL;DR: In this paper, the authors studied those Poisson structures for which the explicit methods of Fedosov can be applied, namely the Poisson structure coming from symplectic Lie algebroids, as well as holomorphic symplectic structures.
Abstract: Recently Kontsevich solved the classification problem for deformation quantizations of all Poisson structures on a manifold In this paper we study those Poisson structures for which the explicit methods of Fedosov can be applied, namely the Poisson structures coming from symplectic Lie algebroids, as well as holomorphic symplectic structures For deformations of these structures we prove the classification theorems and a general a general index theorem
115 citations
TL;DR: In this paper, the authors present a K-theoretic approach to the Guillemin-Sternberg conjecture about the commutativity of geometric quantization and symplectic reduction.
112 citations
TL;DR: In this paper, a projective symplectic manifold is defined as a Kaehler manifold of even dimension n with a non-degenerate holomorphic 2-form ω, i.e. ω is a nowhere vanishing n-form.
Abstract: By a symplectic manifold (or a symplectic n-fold) we mean a compact Kaehler manifold of even dimension n with a non-degenerate holomorphic 2form ω, i.e. ω is a nowhere-vanishing n-form. This notion is generalized to a variety with singularities. We call X a projective symplectic variety if X is a normal projective variety with rational Gorenstein singularities and if the regular locus U of X admits a non-degenerate holomorphic 2-form ω. A symplectic variety will play an important role together with a singular Calabi-Yau variety in the generalized Bogomolov decomposition conjecture. Now that essentially a few examples of symplectic manifolds are discovered, it seems an important task to seek new symplectic manifolds by deforming symplectic varieties. In this paper we shall study a projective symplectic variety from a view point of deformation theory. If X has a resolution π : X → X such that (X, πω) is a symplectic manifold, we say that X has a symplectic resolution. Our first results are concerned with a birational contraction map of a symplectic manifold.
110 citations
Book•
23 Nov 2001
TL;DR: In this paper, the authors show how to construct periodic orbits for Tn x IRn invariant Hamiltonian systems using twist maps of the annulus and the Aubry-Mather theorem generating phases.
Abstract: Twist maps of the annulus the Aubry-Mather theorem ghost circles symplectic twist maps periodic orbits for symplectic twist maps of Tn x IRn invariant manifolds Hamiltonian systems vs. twist maps periodic orbits for Hamiltonian systems generalizations of the Aubry-Mather theorem generating phases and symplectic topology.
103 citations
TL;DR: In this article, the generalized rational blowdown, a surgery on smooth 4-manifolds, can be performed in the symplectic category and is shown to work well on smooth rectangles.
Abstract: We prove that the generalized rational blowdown, a surgery on smooth 4-manifolds, can be performed in the symplectic category. AMS Classication 57R17; 57R15, 57M50
72 citations
TL;DR: In this paper, a complete set of invariants for such spaces when they are "centered" and the moment map is proper is provided, which is an important step towards global classification.
Abstract: We consider symplectic manifolds with Hamiltonian torus actions which are “almost but not quite completely integrable”: the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are “centered” and the moment map is proper. In particular, this classifies the preimages under the moment map of all sufficiently small open sets, which is an important step towards global classification. As an application, we construct a full packing of each of the Grassmannians Gr(2,R5) and Gr(2,R6) by two equal symplectic balls.
69 citations
TL;DR: In this paper, the authors consider a generalization of manifolds and orbifolds called quasifolds, which are locally isomorphic to the quotient of the space R k by the action of a discrete group, typically they are not Hausdorff topological spaces.
69 citations
TL;DR: In this paper, the composition of arrows in integrable Poisson manifolds is interpreted as a composition of isomorphism classes of dual pairs, with symplectic groupoids as units.
Abstract: Symplectic reduction is reinterpreted as the composition of arrows in the category of integrable Poisson manifolds, whose arrows are isomorphism classes of dual pairs, with symplectic groupoids as units. Morita equivalence of Poisson manifolds amounts to isomorphism of objects in this category.
Posted Content•
TL;DR: In this paper, it was shown that the singular locus of a symplectic singularity has no codimension 3 irreducible components and that it is terminal if and only if it has codimensions ≥ 4.
Abstract: In this paper we shall prove that the singular locus of a symplectic singularity has no codimension 3 irreducible components. As a corollary, a symplectic singularity is terminal if and only if its singular locus has codimension $\geq 4$.
It is hoped that a symplectic singularity has much stronger properties.
TL;DR: In this article, a unified theory for discrete and differential linear Hamiltonian systems on an arbitrary time scale T was proposed, where disconjugacy for conjoined bases of a dynamic system is defined and proved to be equivalent to the positivity of the associated quadratic functional.
Abstract: In this paper we study qualitative properties of the so-called symplectic dynamic system (S) z^\Delta=S(t)z on an arbitrary time scale T, providing a unified theory for discrete symplectic systems (T=Z) and differential linear Hamiltonian systems (T=R). We define disconjugacy (no focal points) for conjoined bases of (S) and prove, under a certain minimal normality assumption, that disconjugacy of (S) on the interval under consideration is equivalent to the positivity of the associated quadratic functional. Such statement is commonly called Jacobi condition. We discuss also solvability of the corresponding Riccati matrix equation and transformations. This work may be regarded as a generalization of the results recently obtained by the second author for linear Hamiltonian systems on time scales.
TL;DR: In this article, Donaldson's approximately holomorphic techniques were used to construct a compact isotropic submanifold of a compact symplectic manifold, and the connection with rational convexity results in the Kahler case was discussed.
Abstract: Using Donaldson's approximately holomorphic techniques, we construct symplectic hypersurfaces lying in the complement of any given compact isotropic submanifold of a compact symplectic manifold. We discuss the connection with rational convexity results in the Kahler case and various applications.
TL;DR: In this article, different blow-up constructions on a symplectic orbifold have been studied for the case of a Hamiltonian torus action not necessarily quasi-free, and the wall-crossing theorem of Guillemin and Sternberg has been generalized to the manifold case.
Abstract: In the first part of this paper we study different blow-upconstructions on symplectic orbifolds. Unlike the manifold case,we can define different blow-ups by using different circleactions. In the second part, we use some of these constructions todescribe the behavior of reduced spaces of a Hamiltonian circleaction on a symplectic orbifold, when passing a critical level ofits Hamiltonian function. Using these descriptions, we generalize,in the manifold case, the wall-crossing theorem of Guillemin and Sternberg to the case of a Hamiltonian torus action not necessarily quasi-free and also the Duistermaat–Heckman theorem to intervalsof values of the Hamiltonian function containing critical values.
TL;DR: In this article, it was shown how one can do symplectic reduction for locally conformal symplectic manifolds, especially with an action of a Lie group, which generalizes well-known procedures for symplectic manifold decomposition to the slightly larger class of locally con-tional manifolds.
TL;DR: In this article, a universal symplectic structure for a constrained system can be constructed in the framework of Birkhoffian generalization of Hamiltonian mechanics, which preserves symbiotic character among derivability from a variational principle, Lie algebra and symplectic geometry.
Journal Article•
TL;DR: In this paper, a topological structure called a hyperpencil on a compact 2n-manifold is defined, motivated by the special case of a linear system of curves on an algebraic manifold.
Abstract: A topological structure is introduced that seems likely to provide a complete topological characterization of compact symplectic manifolds. The article begins with a leisurely introduction to symplectic manifolds from a topological viewpoint. It then focuses on Thurston's construction of a symplectic structure on the total space of a fiber bundle. This is generalized to a technique for putting a symplectic structure on the domain of a J-holomorphic map. A topological structure called a hyperpencil on a compact 2n-manifold is then defined; this is motivated by the special case of a linear system of curves on an algebraic manifold, and it generalizes the notion of a Lefschetz pencil on a 4-manifold (although the critical points of a hyperpencil can be much more complicated). A deformation class of hyperpencils determines an isotopy class of symplectic forms, via the above generalization of Thurston's construction. This correspondence seems to be essentially an inverse to the technique of Donaldson and Auroux for constructing linear systems on symplectic manifolds. The likely end result is that any symplectic form whose cohomology class is rational should be realized up to scale by a hyperpencil. This would topologically characterize symplectic manifolds as being those smooth manifolds admitting hyperpencils, and put a dense subset of all symplectic forms on a manifold (up to scale and isotopy) in bijective correspondence with the set of all hyperpencils on it modulo a suitable equivalence relation.
TL;DR: In this paper, a double compactified D=11 supermembrane with nontrivial wrapping was formulated as a symplectic non-commutative gauge theory on the world volume.
Abstract: It is shown that a double compactified D=11 supermembrane with nontrivial wrapping may be formulated as a symplectic noncommutative gauge theory on the world volume The symplectic noncommutative structure is intrinsically obtained from the symplectic two-form on the world volume defined by the minimal configuration of its Hamiltonian The gauge transformations on the symplectic fibration are generated by the area preserving diffeomorphisms on the world volume Geometrically, this gauge theory corresponds to a symplectic fibration over a compact Riemann surface with a symplectic connection
TL;DR: In this article, a rational quadratic expression invariant to the case of four variables was proposed and all mappings satisfying these conditions were shown to be integrable either as four-dimensional mappings with two explicit integrals which are in involution with respect to the symplectic structure and which can also be inferred from the periodic reductions of the double-discrete versions of the modified Korteweg-deVries (ΔΔMKdV) and sine-Gordon (SG) equations or by reduction to two-dimensionalmappings with
Abstract: We investigate the generalisations of the Quispel, Roberts and Thompson (QRT) family of mappings in the plane leaving a rational quadratic expression invariant to the case of four variables. We assume invariance of the rational expression under a cyclic permutation of variables and we impose a symplectic structure with Poisson brackets of the Weyl type. All mappings satisfying these conditions are shown to be integrable either as four-dimensional mappings with two explicit integrals which are in involution with respect to the symplectic structure and which can also be inferred from the periodic reductions of the double-discrete versions of the modified Korteweg–deVries (ΔΔMKdV) and sine-Gordon (ΔΔsG) equations or by reduction to two-dimensional mappings with one integral of the symmetric QRT family.
TL;DR: In this paper, the construction of toric Kahler metrics on 2n-manifolds with hamiltonian n-torus action is discussed and a simple derivation of the Guillemin formula for a distinguished Kahler metric on any such manifold is presented.
Abstract: We discuss the construction of toric Kahler metrics on
symplectic 2n-manifolds with hamiltonian n-torus
action and present a simple derivation of the Guillemin
formula for a distinguished Kahler metric on any
such manifold. the results also apply to orbifolds.
TL;DR: In this paper, Gromov's non-squeezing theorem can be used to quantize phase space in cells, which leads to the correct energy levels for integrable systems and to Maslov quantization of Lagrangian manifolds.
Abstract: We show that a result of symplectic topology, Gromov's non-squeezing theorem, also known as the `principle of the symplectic camel', can be used to quantize phase space in cells. That quantization scheme leads to the correct energy levels for integrable systems and to Maslov quantization of Lagrangian manifolds by purely topological arguments. We finally show that the argument leading to the proof of the non-squeezing theorem leads to a classical form of Heisenberg's inequalities.
TL;DR: In this paper, Li and Ruan established some relations between Gromov-Witten invariants of a semipositive manifold M and its blow-ups along a smooth surface.
Abstract: In this paper, using the gluing formula of Gromov–Witten invariants for symplectic cutting developed by Li and Ruan, we established some relations between Gromov–Witten invariants of a semipositive symplectic manifold M and its blow-ups along a smooth surface.
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TL;DR: In this paper, the authors show that the convexity theorem, the Kostant multiplicity theorem and the quantization commutes with reduction theorem for circle subgroups of G are basically just theorems about G-actions on graphs.
Abstract: Let G be an n-dimensional torus and $\tau$ a Hamiltonian action of G on a compact symplectic manifold, M. If M is pre-quantizable one can associate with $\tau$ a representation of G on a virtual vector space, Q(M), by $\spin^{\CC}$-quantization. If M is a symplectic GKM manifold we will show that several well-known theorems about this ``quantum action'' of G: for example, the convexity theorem, the Kostant multiplicity theorem and the ``quantization commutes with reduction'' theorem for circle subgroups of G, are basically just theorems about G-actions on graphs.
TL;DR: In this paper, a simple construction yielding homology classes in (non-simply-connected) symplectic four-manifolds which admit infinitely many pairwise non-isotopic symplectic representatives was given.
Abstract: We give a simple construction yielding homology classes in (non-simply-connected) symplectic four-manifolds which admit infinitely many pairwise non-isotopic symplectic representatives. Examples are constructed in which the symplectic curves can have arbitrarily large genus. The examples are built from surface bundles over surfaces and involve only elementary techniques. As a corollary we see that a blow-up of any simply-connected complex projective surface contains a connected symplectic surface not isotopic to any complex curve.
TL;DR: In this article, a complete classification of symplectic filiform complex Lie algebras and a description of all their symplectic structures in dimension ≤ 10 is given, and a classification of all the corresponding structures in the manifold is given.
Journal Article•
TL;DR: In this article, a new construction of maps from a compact manifold of any dimension to CP2 and the associated monodromy invariants is presented. But it is not shown how to construct such a map from a manifold to a set of words in braid groups.
Abstract: After reviewing recent results on symplectic Lefschetz pencils and symplectic branched covers of \CP2, we describe a new construction of maps from symplectic manifolds of any dimension to \CP2 and the associated monodromy invariants. We also show that a dimensional induction process makes it possible to describe any compact symplectic manifold by a series of words in braid groups and a word in a symmetric group.
TL;DR: In this paper, it was shown that if a Lie group acts properly on a co-oriented contact manifold preserving the contact structure, then the contact quotient is topologically a stratified space.
Abstract: We show that if a Lie group acts properly on a co-oriented contact manifold preserving the contact structure, then the contact quotient is topologically a stratified space (in the sense that a neighborhood of a point in the quotient is a product of a disk with a cone on a compact stratified space). As a corollary, we obtain that symplectic quotients for proper Hamiltonian actions are topologically stratified spaces in this strong sense thereby extending and simplifying previous work.