Showing papers on "Symplectic vector space published in 2002"

Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds

Abstract: In their work on symplectic manifolds, Donaldson and Auroux use analogues of holomorphic sections of an ample line bundle L over a symplectic manifold M to create symplectically embedded zero sections and almost holomorphic maps to various spaces. Their analogues were termed ‘asymptotically holomorphic’ sequences {sN} of sections of L . We study another analogue H J (M, L) of holomorphic sections, which we call ‘almostholomorphic’ sections, following a method introduced earlier by Boutet de Monvel Guillemin [BoGu] in a general setting of symplectic cones. By definition, sections in H J (M, L ) lie in the range of a Szegö projector ΠN . Starting almost from scratch, and only using almostcomplex geometry, we construct a simple parametrix for ΠN of precisely the same type as the Boutet de Monvel-Sjöstrand parametrix in the holomorphic case [BoSj]. We then show that ΠN (x, y) has precisely the same scaling asymptotics as does the holomorphic Szegö kernel as analyzed in [BSZ1]. The scaling asymptotics imply more or less immediately a number of analogues of well-known results in the holomorphic case, e.g. a Kodaira embedding theorem and a Tian almost-isometry theorem. We also explain how to modify Donaldson’s constructions to prove existence of quantitatively transverse sections in H J (M, L ).

Four dimensions from two in symplectic topology

Abstract: This paper introduces two-dimensional diagrams that are slight general- izations of moment map images for toric four-manifolds and catalogs techniques for reading topological and symplectic properties of a symplectic four-manifold from these diagrams. The paper offers a purely topological approach to toric manifolds as well as methods for visualizing other manifolds, including the K3 surface, and for visualizing certain surgeries. Most of the techniques extend to higher dimensions.

Symplectic Conifold Transitions

Abstract: We introduce a symplectic surgery in six dimensions which collapses Lagrangian three-spheres and replaces them by symplectic two-spheres. Under mirror symmetry it corresponds to an operation on complex 3-folds studied by Clemens, Friedman and Tian. We describe several examples which show that there are either many more Calabi-Yau manifolds (e.g., rigid ones) than previously thought or there exist "symplectic Calabi-Yaus" — non-Kahler symplectic 6-folds with c1 = 0. The analogous surgery in four dimensions, with a generalisation to ADE-trees of Lagrangians, implies that the canonical class of a minimal complex surface contains symplectic forms if and only if it has positive square.

On symplectic leaves and integrable systems in standard complex semisimple Poisson-Lie groups

Abstract: We provide an explicit description of symplectic leaves of a simply connected connected semisimple complex Lie group equipped with the standard Poisson-Lie structure. This sharpens previously known descriptions of the symplectic leaves as connected components of certain varieties. Our main tool is the machinery of twisted generalized minors. They also allow us to present several quasi-commuting coordinate systems on every symplectic leaf. As a consequence, we construct new completely integrable systems on some special symplectic leaves.

Symplectic Shifted Tableaux and Deformations of Weyl's Denominator Formula for sp (2 n )

Abstract: A determinantal expansion due to Okada is used to derive both a deformation of Weyl's denominator formula for the Lie algebra sp(2n) of the symplectic group and a further generalisation involving a product of the deformed denominator with a deformation of flagged characters of sp(2n). In each case the relevant expansion is expressed in terms of certain shifted sp(2n)-standard tableaux. It is then re-expressed, first in terms of monotone patterns and then in terms of alternating sign matrices.

The Hamilton-Jacobi method and Hamiltonian maps

Abstract: A method for constructing time-step-based symplectic maps for a generic Hamiltonian system subjected to perturbation is developed. Using the Hamilton–Jacobi method and Jacobi’s theorem in finite periodic time intervals, the general form of the symplectic maps is established. The generating function of the map is found by the perturbation method in the finite time intervals. The accuracy of the maps is studied for fully integrable and partially chaotic Hamiltonian systems and compared to that of the symplectic integration method.

A symplectic slice theorem

Abstract: We provide a model for an open invariant neighborhood of any orbit in a symplectic manifold endowed with a canonical proper symmetry. Our results generalize the constructions of Marle and Guillemin and Sternberg for canonical symmetries that have an associated momentum map. In these papers the momentum map played a crucial role in the construction of the tubular model. The present work shows that in the construction of the tubular model, the so-called Chu map, can be used instead, which exists for any canonical action, unlike the momentum map. Hamilton's equations for any invariant Hamiltonian function take on a particularly simple form in these tubular variables. As an application we will find situations, that we will call tubewise Hamiltonian, in which the existence of a standard momentum map in invariant neighborhoods is guaranteed.

Geometry of Symplectic Intersections

Abstract: In this paper we survey several intersection and non-intersection phenom­ ena appearing in the realm of symplectic topology. We discuss their im­ plications and finally outline some new relations of the subject to algebraic geometry.

The generic symplectic C^{1}-diffeomorphisms of four-dimensional symplectic manifolds are hyperbolic, partially hyperbolic or have a completely elliptic periodic point

Abstract: We prove that if (M,\omega) is a connected and compact four-dimensional symplectic manifold, there exist three open sets U_1 , U_2 , U_3 of {\rm Diff}^1_{\omega}(M) (for the C^1 topology) such that: U_1\cup U_2\cup U_3 is dense in {\rm Diff}^1_{\omega}(M) ; f\in U_1 if and only if f is Anosov and transitive; f\in U_2 if and only if f is partially hyperbolic; and f\in U_3 if and only if f has a stable completely elliptic periodic point.

The 'symplectic camel principle' and semiclassical mechanics

Abstract: We propose a theory of semiclassical mechanics in phase space based on the notion of quantized symplectic area. The definition of symplectic area makes use of a deep topological property of symplectic mappings, known as the 'principle of the symplectic camel' which places stringent conditions on the global geometry of Hamiltonian mechanics. Following this principle, symplectic mappings—and hence Hamiltonian flows—are much more rigid than Liouville's theorem suggests. The dynamical objects of our semiclassical theory are 'waveforms', whose definition requires the notion of square root of de Rham forms. The arguments of these square roots are calculated by using the properties of a generalized Maslov index. The motion of waveforms is determined by Hamiltonian mechanics, and the local expressions of these moving waveforms on configuration space are the usual approximate solutions of WKB-Maslov theory.

Moment maps and symmetric multilinear forms associated with symplectic classes

Abstract: where $ denotes the Lie algebra of G. This /J, is uniquely determined by the G-action on M as above, and is called the reduced moment map. Let us first consider the case where G is a fc-dimensional torus T = (S)* with the associated Lie algebra t. Then the image /x(M) of the moment map is a compact convex polytope (cf. Atiyah [1], Guillemin and Sternberg [14]). The kernel, denoted by tz, of the exponential map exp : t —> T is called the lattice in t, and points in the lattice are called integral points. They in turn define the dual lattice t| and integral points in t*. By setting iq := tz ^z Q and tq := tj ®z Q? we have rational points in t and t*. A convex polytope in t* is said to be integral or rational, according as all vertices are integral points or rational points, respectively. The fixed point set M of the T-action on M sits in the critical point set for fi, and the image /i(M) is a finite subset of t*. The following proposition, which was originally conjectured by Atiyah [1] in the special case of projective algebraic manifolds, plays a key role in our work:

The Symplectic Geometry of Polygons in the 3-Sphere

Abstract: We study the symplectic geometry of the moduli spaces Mr= Mr(S3) of closed n-gons with fixed side-lengths in the 3-sphere. We prove that these moduli spaces have symplectic structures ob- tained by reduction of the fusion product of n conjugacy classes in SU(2) by the diagonal conjugation action of SU(2). Here the fusion product of nconjugacy classes is a Hamiltonian quasi-Poisson SU(2)- manifold in the sense of (AKSM). An integrable Hamiltonian system is constructed on Mr in which the Hamiltonian flows are given by bending polygons along a maximal collection of nonintersecting diagonals. Finally, we show the symplectic structure on Mr relates to the symplectic structure obtained from gauge-theoretic description of Mr. The results of this paper are analogues for the 3-sphere of results obtained for Mr(H 3 ), the moduli space of n-gons with fixed side-lengths in hyperbolic 3-space (KMT), and for Mr(E 3 ), the moduli space of n-gons with fixed side-lengths in E 3 (KM1).

The moduli space of complex Lagrangian submanifolds

Abstract: Following an earlier paper on the differential-geometric structure of the moduli space of special Lagrangian submanifolds in a Calabi-Yau manifold, we follow an analogous approach for compact complex Lagrangian submanifolds of a (K\"ahlerian) complex symplectic manifold. The natural geometric structure on the moduli space is a special K\"ahler metric, but we offer a different point of view on the local differential geometry of these, based on the structure of a submanifold of $V\times V$ (where $V$ is a symplectic vector space) which is Lagrangian with respect to two constant symplectic forms. As an application, we show using this point of view how the hyperk\"ahler metric of Cecotti, Ferrara and Girardello associated to a special K\"ahler structure fits into the Legendre transform construction of Lindstr\"om and Ro\v cek.

Poisson deformations of symplectic quotient singularities

Abstract: We establish a connection between smooth symplectic resolutions and symplectic deformations of a (possibly singular) affine Poisson variety. In particular, let V be a finite-dimensional complex symplectic vector space and G\subset Sp(V) a finite subgroup. Our main result says that the so-called Calogero-Moser deformation of the orbifold V/G is, in an appropriate sense, a versal Poisson deformation. That enables us to determine the algebra structure on the rational cohomology H^*(X) of any smooth symplectic resolution X \to V/G (multiplicative McKay correspondence). We prove further that if G is an irreducible Weyl group in GL(h) and V=h+ h^* then no smooth symplectic resolution of V/G exists unless G is of types A,B, or C.

Seiberg-Witten Theory, Symplectic Forms, and Hamiltonian Theory of Solitons

Abstract: This is an expanded version of lectures given in Hangzhou and Beijing, on the symplectic forms common to Seiberg-Witten theory and the theory of solitons Methods for evaluating the prepotential are discussed The construction of new integrable models arising from supersymmetric gauge theories are reviewed, including twisted Calogero-Moser systems and spin chain models with twisted monodromy conditions A practical framework is presented for evaluating the universal symplectic form in terms of Lax pairs A subtle distinction between a Lie algebra and a Lie group version of this symplectic form is clarified, which is necessary in chain models

Rational surfaces and symplectic 4-manifolds with one basic class

Abstract: We present constructions of simply connected symplectic 4-manifolds which have (up to sign) one basic class and which fill up the geographical region between the half-Noether and Noether lines.

Coherent State Projection Operator Representation of Symplectic Transformations as a Loyal Representation of Symplectic Group

Abstract: We find that the coherent state projection operator representation of symplectic transformation constitutes a loyal group representation of symplectic group. The result of successively applying squeezing operators on number state can be easily derived.

Fedosov quantization on symplectic ringed spaces

Abstract: We expose the basics of the Fedosov quantization procedure, placed in the general framework of symplectic ringed spaces. This framework also includes some Poisson manifolds with nonregular Poisson structures, presymplectic manifolds, complex analytic symplectic manifolds, etc.

Seiberg-Witten theory, symplectic forms, and Hamiltonian theory of solitons

Abstract: This is an expanded version of lectures given in Hangzhou and Beijing, on the symplectic forms common to Seiberg-Witten theory and the theory of solitons. Methods for evaluating the prepotential are discussed. The construction of new integrable models arising from supersymmetric gauge theories are reviewed, including twisted Calogero-Moser systems and spin chain models with twisted monodromy conditions. A practical framework is presented for evaluating the universal symplectic form in terms of Lax pairs. A subtle distinction between a Lie algebra and a Lie group version of this symplectic form is clarified, which is necessary in chain models.

On certain canonical diffeomorphisms in symplectic and Poisson geometry

Abstract: The canonical involution of a double (=iterated) tangent bundle may be dualized in different ways to yield relations between the Tulczyjew diffeomorphism, the Poisson anchor associated with the standard symplectic structure on the cotangent space,and the reversal diffeomorphism. We show that the constructions which yield these maps extend very generally to the double Lie algebroids of double Lie groupoids, where they play a crucial role in the relations between double Lie algebroids and Lie bialgebroids.

Actions symplectiques de groupes compacts (Symplectic Actions of Compact Groups)

Abstract: For any symplectic action of a compact connected group on a compact connected symplectic manifold, we show that the intersection of the Weyl chamber with the image of the moment map is a closed convex polyhedron. This extends Atiyah–Guillemin–Sternberg–Kirwan's convexity theorems to non-Hamiltonian actions. As a consequence, we describe those symplectic actions of a torus which are coisotropic (or multiplicity free), i.e. which have at least one coisotropic orbit: they are the product of an Hamiltonian coisotropic action by an anhamiltonian one. The Hamiltonian coisotropic actions have already been described by Delzant thanks to the convex polyhedron. The anhamiltonian coisotropic actions are actions of a central torus on a symplectic nilmanifold. This text is written as an introduction to the theory of symplectic actions of compact groups since complete proofs of the preliminary classical results are given.

Nonrational, nonsimple convex polytopes in symplectic geometry

Abstract: In this research announcement we associate to each convex polytope, possibly nonrational and nonsimple, a family of compact spaces that are stratified by quasifolds, i.e., the strata are locally modeled by Rk modulo the action of a discrete, possibly infinite, group. Each stratified space is endowed with a symplectic structure and a moment mapping having the property that its image gives the original polytope back. These spaces may be viewed as a natural generalization of symplectic toric varieties to the nonrational setting. We provide here the explicit construction of these spaces, and a thorough description of the stratification.

Hypersurfaces in symplectic affine geometry

Abstract: We study properties of hypersurfaces of the standard symplectic space ( R 2n ,ω) , which are invariant under affine symplectic transformations. In this framework, we describe the invariants of hypersurfaces and discuss the existence of an isoperimetric inequality.

Index functions for symplectic paths

Abstract: In this chapter, we introduce an index function theory for paths in the symplectic groups started from the identity, i.e., elements in Pτ(2n) with τ > 0 defined by (2.0.1): . For τ> 0 and ω∈U, we further define the set of ω-non-degenerate paths by (1) and the set of ω-degenerate paths by (2) .