Showing papers on "Symplectic vector space published in 2005"

New aspects of the ddc-lemma

Abstract: We produce examples of generalized complex structures on manifolds by generalizing results from symplectic and complex geometry. We produce generalized complex structures on symplectic fibrations over a generalized complex base. We study in some detail different invariant generalized complex structures on compact Lie groups and provide a thorough description of invariant structures on nilmanifolds, achieving a classification on 6-nilmanifolds. We study implications of the `dd^c-lemma' in the generalized complex setting. Similarly to the standard dd^c-lemma, its generalized version induces a decomposition of the cohomology of a manifold and causes the degeneracy of the spectral sequence associated to the splitting d = \del + \delbar at E_1. But, in contrast with the dd^c-lemma, its generalized version is not preserved by symplectic blow-up or blow-down (in the case of a generalized complex structure induced by a symplectic structure) and does not imply formality.

Singular Lefschetz pencils

Abstract: We consider structures analogous to symplectic Lefschetz pencils in the context of a closed 4–manifold equipped with a “near-symplectic” structure (ie, a closed 2–form which is symplectic outside a union of circles where it vanishes transversely). Our main result asserts that, up to blowups, every near-symplectic 4–manifold (X, ω) can be decomposed into (a) two symplectic Lefschetz fibrations over discs, and (b) a fibre bundle over S 1 which relates the boundaries of the Lefschetz fibrations to each other via a sequence of fibrewise handle additions taking place in a neighbourhood of the zero set of the 2–form. Conversely, from such a decomposition one can recover a near-symplectic structure.

Generalized calabi–yau structures, k3 surfaces, and b-fields

Abstract: Generalized Calabi–Yau structures, a notion recently introduced by Hitchin, are studied in the case of K3 surfaces. We show how they are related to the classical theory of K3 surfaces and to moduli spaces of certain SCFT as studied by Aspinwall and Morrison. It turns out that K3 surfaces and symplectic structures are both special cases of this general notion. The moduli space of generalized Calabi–Yau structures admits a canonical symplectic form with respect to which the moduli space of symplectic structures is Lagrangian. The standard theory of K3 surfaces implies surjectivity of the period map and a weak form of the Global Torelli theorem.

Symplectic Geometry and Hilbert's Fourth Problem

Abstract: Inspired by Hofer's definition of a metric on the space of compactly supported Hamiltonian maps on a symplectic manifold, this paper exhibits an area-length duality between a class of metric spaces and a class of symplectic manifolds. Using this duality, it is shown that there is a twistor-like correspondence between Finsler metrics on ℝPn whose geodesics are projective lines and a class of symplectic forms on the Grassmannian of 2-planes in ℝn+1.

Formal Symplectic Groupoid of a Deformation Quantization

Abstract: We give a self-contained algebraic description of a formal symplectic groupoid over a Poisson manifold M. To each natural star product on M we then associate a canonical formal symplectic groupoid over M. Finally, we construct a unique formal symplectic groupoid ‘with separation of variables’ over an arbitrary Kahler-Poisson manifold.

The Gromov-Witten potential associated to a TCFT

Abstract: This is the sequel to my preprint "TCFTs and Calabi-Yau categories", mathQA/0412149 Here we extend the results of that paper to construct, for certain Calabi-Yau A-infinity categories, something playing the role of the Gromov-Witten potential This is a state in the Fock space associated to periodic cyclic homology, which is a symplectic vector space Applying this to a suitable A-infinity version of the derived category of sheaves on a Calabi-Yau yields the B model potential, at all genera The construction doesn't go via the Deligne-Mumford spaces, but instead uses the Batalin-Vilkovisky algebra constructed from the uncompactified moduli spaces of curves by Sen and Zwiebach The fundamental class of Deligne-Mumford space is replaced here by a certain solution of the quantum master equation, essentially the "string vertices" of Zwiebach On the field theory side, the BV operator has an interpretation as the quantised differential on the Fock space for periodic cyclic chains Passing to homology, something satisfying the master equation yields an element of the Fock space

Representations of two-parameter quantum orthogonal and symplectic groups

Abstract: We investigate the finite-dimensional representation theory of two- parameter quantum orthogonal and symplectic groups that we found in (BGH) under the assumption that rs −1 is not a root of unity and extend some results (BW1, BW2) obtained for type A to types B, C and D. We construct the corresponding R-matrices and the quantum Casimir operators, by which we prove that the complete reducibility Theorem also holds for the categories of finite-dimensional weight modules for types B, C, D.

On theoretical and numerical aspects of symplectic Gram Schmidt-like algorithms

Abstract: Gram–Schmidt-like orthogonalization process with respect to a given skew-symmetric scalar product is a key step in model reduction methods, structure-preserving, for large sparse Hamiltonian eigenvalue problem. Theoretical as well as numerical aspects of this step do not benefit of enough attention, compared to the one allowed to the classical Gram–Schmidt algorithm and its modified version. The aim of this paper is to revisit the symplectic Gram–Schmidt algorithms, to built some modified versions and to deal with their theoretical and numerical features.

Minimality and irreducibility of symplectic four-manifolds

Abstract: We prove that all minimal symplectic four-manifolds are essentially irreducible. We also clarify the relationship between holomorphic and symplectic minimality of K\"ahler surfaces. This leads to a new proof of the deformation-invariance of holomorphic minimality for complex surfaces with even first Betti number which are not Hirzebruch surfaces.

Complex symplectic spaces and boundary value problems

Abstract: This paper presents a review and summary of recent research on the boundary value problems for linear ordinary and partial differential equations, with special attention to the investigations of the current authors emphasizing the applications of complex symplectic spaces. In the first part of the previous century, Stone and von Neumann formulated the theory of self-adjoint extensions of symmetric linear operators on a Hilbert space; in this connection Stone developed the properties of self-adjoint differential operators generated by boundary value problems for linear ordinary differential equations. Later, in diverse papers, Glazman, Krein and Naimark introduced certain algebraic techniques for the treatment of appropriate generalized boundary conditions. During the past dozen years, in a number of monographs and memoirs, the current authors of this expository summary have developed an extensive algebraic structure, complex symplectic spaces, with applications to both ordinary and partial linear boundary value problems. As a consequence of the use of complex symplectic spaces, the results offer new insights into the theory and use of indefinite inner product spaces, particularly Krein spaces, from an algebraic viewpoint. For instance, detailed information is obtained concerning the separation and coupling of the boundary conditions at the endpoints of the intervals for ordinary differential operators (see the Balanced Intersection Principle), and the introduction of the generalized boundary conditions over the region for some elliptic partial differential operators (see the Harmonic operator).

A de Rham theorem for symplectic quotients

Abstract: We introduce a de Rham model for stratified spaces arising from symplectic reduction. It turns out that the reduced symplectic form and its powers give rise to well-defined cohomology classes, even on a singular symplectic quotient.

Packing symplectic manifolds by hand

Abstract: We construct explicit maximal symplectic packings of minimal rational and ruled symplectic 4-manifolds by few balls in a very simple way.

Locally holomorphic maps yield symplectic structures

Abstract: For a smooth map $f:X^4\to\Sigma^2$ that is locally modeled by holomorphic maps, the domain is shown to admit a symplectic structure that is symplectic on some regular fiber, if and only if $f^*[\Sigma] e0$. If so, the space of symplectic forms on $X$ that are symplectic on all fibers is nonempty and contractible. The cohomology classes of these forms vary with the maximum possible freedom on the reducible fibers, subject to the obvious constraints. The above results are derived via an analogous theorem for locally holomorphic maps $f:X^{2n}\to Y^{2n-2}$ with $Y$ symplectic.

Symplectic torus bundles and group extensions

Abstract: Symplectic torus bundles ξ : T 2 → E → B are classified by the second cohomology group of B with local coefficients H1(T 2). For B a com- pact, orientable surface, the main theorem of this paper gives a necessary and sufficient condition on the cohomology class corresponding to ξ for E to admit a symplectic structure compatible with the symplectic bundle structure of ξ: namely, that it be a torsion class. The proof is based on a group-extension- theoretic construction of J. Huebschmann, 1981. A key ingredient is the notion of fibrewise-localization.

Enlarging the Hamiltonian group

Abstract: This paper investigates ways to enlarge the Hamiltonian subgroup Ham of the symplectomorphism group Symp(M) of the symplectic manifold (M,ω) to a group that both intersects every connected component of Symp(M) and characterizes symplectic bundles with fiber M and closed connection form. As a consequence, it is shown that bundles with closed connection form are stable under appropriate small perturbations of the symplectic form. Further, the manifold (M,ω) has the property that every symplectic M -bundle has a closed connection form if and only if the flux group vanishes and the flux homomorphism extends to a crossed homomorphism defined on the whole group Symp(M). The latter condition is equivalent to saying that a connected component of the commutator subgroup [Symp,Symp] intersects the identity component of Symp only if it also intersects Ham. It is not yet clear when this condition is satisfied. We show that if the symplectic form vanishes on 2-tori the flux homomorphism extends to the subgroup of Symp acting trivially on π1(M). We also give an explicit formula for the Kotschick–Morita extension of Flux in the monotone case. The results in this paper belong to the realm of soft symplectic topology, but raise some questions that may need hard methods to answer.

Convexity of Multi-valued Momentum Maps

Abstract: A famous theorem of Atiyah, Guillemin and Sternberg states that, given a Hamiltonian torus action, the image of the momentum map is a convex polytope. We prove that this result can be extended to the case in which the action is non-Hamiltonian. Our generalization of the theorem states that, given a symplectic torus action, the momentum map can be defined on an appropriate covering of the manifold and its image is the product of a convex polytope along a rational subspace times the orthogonal vector space. We also prove that this decomposition in direct product is stable under small equivariant perturbations of the symplectic structure; this, in particular, means that the property of being Hamiltonian is locally stable. The technique developed allows us to extend the result to any compact group action and also to deduce that any symplectic n-torus action, with fixed points, on a compact 2n-dimensional manifold, is Hamiltonian.

Construction of Ricci-type connections by reduction and induction

Abstract: Given the Euclidean space ℝ2n+2 endowed with a constant symplectic structure and the standard flat connection, and given a polynomial of degree 2 on that space, Baguis and Cahen [1] have defined a reduction procedure which yields a symplectic manifold endowed with a Ricci-type connection. We observe that any symplectic manifold (M, ω) of dimension 2n (n ≥ 2) endowed with a symplectic connection of Ricci type is locally given by a local version of such a reduction.

Twisted symplectic reflection algebras

Abstract: In this paper we introduce the notion of twisted symplectic reflection algebras and describe the category of representations of such an algebra associated to a non-faithful G-action in terms of those for faithful actions of G.

A Z_2-orbifold model of the symplectic fermionic vertex operator superalgebra

Abstract: We construct an irrational C_2-cofinite vertex operator algebra associatted to a finite dimensional vector space with a nondegenerate skew-symmetric bilinear form. We also classify its equivalence classes of irreducible modules and determine its automorphism group.

Contact Path Geometries

Abstract: Contact path geometries are curved geometric structures on a contact manifold comprising smooth families of paths modeled on the family of all isotropic lines in the projectivization of a symplectic vector space. Locally such a structure is equivalent to the graphs in the space of independent and depedent variables of the family of solutions of a system of an odd number of second order ODE's subject to a single maximally non-integrable constraint. A subclass of contact path geometries is distinguished by the vanishing of an invariant contact torsion. For this subclass the equivalence problem is solved by constructing a normalized Cartan connection using the methods of Tanaka-Morimoto-Cap-Schichl. The geometric meaning of the contact torsion is described. If a secondary contact torsion vanishes then the locally defined space of contact paths admits a split quaternionic contact structure (analogous to the quaternionic contact structures studied by O. Biquard).

Existence of symplectic structures on torus bundles over surfaces

Abstract: Let E be the total space of a locally trivial torus bundle over a surface Σg of genus g > 1. Using Seiberg–Witten theory and spectral sequences, we prove that E carries a symplectic structure if and only if the homology class of the fiber [T2] is nonzero in H2(E, ℝ).

New symplectic 4–manifolds with b + =1

Abstract: In 1995 Dusa McDuff and Dietmar Salamon conjectured the existence of symplectic 4–manifolds (X,ω) which satisfy b+=1, K2=0, K·ω>0, and which fail to be of Lefschetz type. This is equivalent to finding a symplectic, homology T2×S2 manifold with nontorsion canonical class and a cohomology ring which is not isomorphic to the cohomology ring of T2×S2. They needed such examples to complete a list of possible symplectic 4–manifolds with b+=1. In that same year Tian-Jun Li and Ai-ko Liu, working from a different point of view, questioned whether there existed symplectic 4–manifolds with b+=1 with Seiberg- Witten invariants that did not depend on the chamber structure of the moduli space. The purpose of this paper is to construct an infinite number of examples which satisfy both requirements.

Geography of symplectic 4-manifolds with Kodaira dimension one

Abstract: The geography problem is usually stated for simply connected symplectic 4-manifolds. When the first cohomology is nontrivial, however, one can restate the problem taking into account how close the symplectic manifold is to satisfying the conclusion of the Hard Lefschetz Theorem, which is measured by a nonnegative integer called the degeneracy. In this paper we include the degeneracy as an extra parameter in the geography problem and show how to fill out the geography of symplectic 4-manifolds with Kodaira dimension 1 for all admissible triples. AMS Classification 57R17; 53D05, 57R57, 57M60

Quantization of Multiply Connected Manifolds

Abstract: The standard (Berezin-Toeplitz) geometric quantization of a compact Kahler manifold is restricted by integrality conditions. These restrictions can be circumvented by passing to the universal covering space, provided that the lift of the symplectic form is exact. I relate this construction to the Baum-Connes assembly map and prove that it gives a strict quantization of the original manifold. I also propose a further generalization, classify the required structure, and provide a means of computing the resulting algebras. These constructions involve twisted group C*-algebras of the fundamental group which are determined by a group cocycle constructed from the cohomology class of the symplectic form. This provides an algebraic counterpart to the Morita equivalence of a symplectic manifold with its fundamental group.

Deformation of integral coisotropic submanifolds in symplectic manifolds

Abstract: In this paper we prove the unobstructedness of the deformation of integral coisotropic submanifolds in symplectic manifolds, which can be viewed as a natural generalization of results of Weinstein for Lagrangian submanifolds.

Homologous non-isotopic symplectic tori in a k3 surface

Abstract: For each member of an infinite family of homology classes in the K3 surface E(2), we construct infinitely many non-isotopic symplectic tori representing this homology class. This family has an infinite subset of primitive classes. We also explain how these tori can be non-isotopically embedded as homologous symplectic submanifolds in many other symplectic 4-manifolds including the elliptic surfaces E(n) for n > 2.