Showing papers on "Symplectic vector space published in 2007"

Contact Geometry and Nonlinear Differential Equations

Abstract: Introduction Part I. Symmetries and Integrals: 1. Distributions 2. Ordinary differential equations 3. Model differential equations and Lie superposition principle Part II. Symplectic Algebra: 4. Linear algebra of symplectic vector spaces 5. Exterior algebra on symplectic vector spaces 6. A Symplectic classification of exterior 2-forms in dimension 4 7. Symplectic classification of exterior 2-forms 8. Classification of exterior 3-forms on a 6-dimensional symplectic space Part III. Monge-Ampere Equations: 9. Symplectic manifolds 10. Contact manifolds 11. Monge-Ampere equations 12. Symmetries and contact transformations of Monge-Ampere equations 13. Conservation laws 14. Monge-Ampere equations on 2-dimensional manifolds and geometric structures 15. Systems of first order partial differential equations on 2-dimensional manifolds Part IV. Applications: 16. Non-linear acoustics 17. Non-linear thermal conductivity 18. Meteorology applications Part V. Classification of Monge-Ampere Equations: 19. Classification of symplectic MAEs on 2-dimensional manifolds 20. Classification of symplectic MAEs on 2-dimensional manifolds 21. Contact classification of MAEs on 2-dimensional manifolds 22. Symplectic classification of MAEs on 3-dimensional manifolds.

Rigid subsets of symplectic manifolds

Abstract: We show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the works of P.Albers and P.Biran-O.Cornea), as well as certain, possibly singular, sets defined in terms of Poisson-commutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semi-simplicity of the quantum homology of symplectic manifolds. The proofs are based on the Floer-theoretical machinery of partial symplectic quasi-states.

Quantitative symplectic geometry

Abstract: A symplectic manifold (M, ω) is a smooth manifold M endowed with a nondegenerate and closed 2-form ω. By Darboux’s Theorem such a manifold looks locally like an open set in some ℝ2n≅ ℂnwith the standard symplectic form and so symplectic manifolds have no local invariants. This is in sharp contrast to Riemannian manifolds, for which the Riemannian metric admits various curvature invariants. Symplectic manifolds do however admit many global numerical invariants, and prominent among them are the so-called symplectic capacities. Symplectic capacities were introduced in 1990 by I. Ekeland and H. Hofer (although the first capacity was in fact constructed by M. Gromov). Since then, lots of new capacities have been defined and they were further studied in. Surveys on symplectic capacities are. Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of symplectic geometry and Hamiltonian dynamics. This is illustrated in Section 2, where we discuss some examples of symplectic capacities and describe a few consequences of their existence. In Section 3 we present an attempt to better understand the space of all symplectic capacities, and discuss some further general properties of symplectic capacities.

Symplectic reflection algebras

Abstract: We survey recent results on the representation theory of symplectic reflection algebras, focusing particularly on connections with symplectic quotient singularities and their resolutions, spaces of representations of quivers, and on category O.

The role of string topology in symplectic field theory

Abstract: We outline a program for incorporating holomorphic curves with Lagrangian boundary conditions into symplectic field theory, with an emphasis on ideas, geometric intuition, and a description of the resulting algebraic structures.

Symplectic quasi-states and semi-simplicity of quantum homology

Abstract: We review and streamline our previous results and the results of Y.Ostrover on the existence of Calabi quasi-morphisms and symplectic quasi-states on symplectic manifolds with semi-simple quantum homology. As an illustration, we discuss the case of symplectic toric Fano 4-manifolds. We present also new results due to D.McDuff: she observed that for the existence of quasi-morphisms/quasi-states it suffices to assume that the quantum homology contains a field as a direct summand, and she showed that this weaker condition holds true for one point blow-ups of non-uniruled symplectic manifolds.

Fedosov quantization in positive characteristic

Abstract: The so-called deformation quantization of symplectic manifolds originally ap peared in C?? symplectic geometry; the crucial breakthrough was independently made in the early 1980s by B. Fedosov and M. De Wilde-P. Lecomte. The input of a deformation quantization problem is a symplectic (or, more generally, a Poisson) manifold M; the output is a non-commutative one-parameter deformation of the algebra of functions on M. Both Fedosov and De Wilde-Lecomte provided general procedures which solve the deformation quantization problem for C?? manifolds. Recently, motivated in part by M. Kontsevich [K], there has been much interest in generalizing the deformation quantization procedures to the case of algebraic manifolds equipped with an algebraic symplectic form (see, e.g., [BK1], [Y], and an earlier paper [NT] for the holomorphic situation). In particular, in [BK1] it has been shown that under some mild assumptions on the manifold, the Fedosov quantization can be made to work in the algebraic setting. The present paper is a continuation of [BK1] (to which we refer the reader for a more complete bibliography and historical discussion). Namely, one of the most important assumptions in [BK1] (as well as in other papers on the subject) was that the field of definition for all algebraic manifolds has characteristic 0. In this paper, we study what happens in the case of positive characteristic. The most obvious new feature of the theory in positive characteristic is the presence of a large Poisson center in the sheaf Ox of functions on a Poisson manifold X: since for any two local functions f,g Ox and any Poisson bracket { ?,?} on Ox we have {fp,g} = 0, the image ?px C Ox of the Frobenius map lies in the center of any Poisson structure. This phenomenon, already observed in [BMR], allows for interesting applications (see, e.g., [BK2]) but makes the quantization procedures more involved. In this paper, we were not able to prove any meaningful results for general quantizations in positive characteristic, and we had to restrict our attention to a special class of them: the so-called Frobenius-constant quantization. Roughly speaking the precise definition is Definition 1.4 below a quantization is Frobenius-constant if the Poisson center Opx C Ox stays central in the quantized algebra Oh For quantization of this type, we were able to achieve, under mild assumptions on the manifold X, a reasonably complete classification theorem. In particular, Frobenius-constant quantizations do exist.

A compact symplectic four-manifold admits only finitely many inequivalent toric actions

Abstract: Let ($M$, \omega) be a four-dimensional compact connected symplectic manifold. We prove that ($M$, \omega) admits only finitely many inequivalent Hamiltonian effective 2-torus actions. Consequently, if $M$ is simply connected, the number of conjugacy classes of 2-tori in the symplectomorphism group Sympl($M$, \omega) is finite. Our proof is “soft”. The proof uses the fact that if a symplectic four-manifold admits a toric action, then the restriction of the period map to the set of exceptional homology classes is proper. In an appendix, we present the Gromov–McDuff properness result for a general compact symplectic four-manifold.

Harish–Chandra homomorphisms and symplectic reflection algebras for wreath-products

Abstract: The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreath-product in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existence of such a construction has been conjectured in [EG]. We also present a new approach to reflection functors and shift functors for generalized preprojective algebras and symplectic reflection algebras associated with wreath-products.

Odd symplectic flag manifolds

Abstract: We define the odd symplectic Grassmannians and flag manifolds, which are smooth projective varieties equipped with an action of the odd symplectic group, analogous to the usual symplectic Grassmannians and flag manifolds. Contrary to the latter, which are the flag manifolds of the symplectic group, the varieties we introduce are not homogeneous. We argue nevertheless that in many respects the odd symplectic Grassmannians and flag manifolds behave like homogeneous varieties; in support of this claim, we compute the automorphism group of the odd symplectic Grassmannians and we prove a Borel-Weil-type theorem for the odd symplectic group.

A surgery for generalized complex structures on 4-manifolds

Abstract: We introduce a surgery for generalized complex manifolds whose input is a symplectic 4-manifold containing a symplectic 2-torus with trivial normal bundle and whose output is a 4-manifold endowed with a generalized complex structure exhibiting type change along a 2-torus. Performing this surgery on a K3 surface, we obtain a generalized complex structure on 3 CP^2 # 19 \overline{CP^2}, which has vanishing Seiberg-Witten invariants and hence does not admit complex or symplectic structure.

Examples of non d ω-exact locally conformal symplectic forms

Abstract: We exhibit two three-parameter families of locally conformal symplectic forms on the solvmanifold M n,k considered in [1], and show, using the Hodge-de Rham theory for the Lichnerowicz cohomology that that they are not d ω exact, i.e. their Lichnerowicz classes are non-trivial (Theorem 1). This has several important geometric consequences (Corollary 2, 3). This also implies that the group of automorphisms of the corresponding locally conformal symplectic structures behaves much like the group of symplectic diffeomorphisms of compact symplectic manifolds. We initiate the classification of the local conformal symplectic forms in each 3-parameter family (Theorem 2, Corollary 1). We also show that the first (and) third Lichnerowicz cohomology classes are non-zero (Theorem 3). We observe finally that the manifolds M n,k carry several interesting foliations and Poisson structures.

Special symplectic six-manifolds

Abstract: We classify nilmanifolds with an invariant symplectic half-flat structure. We study the transverse or quotient geometry of six-manifolds with an SU(3)-structure preserved by a Killing vector field, giving characterizations in the symplectic half-flat and integrable case.

On symplectic 4-manifolds with prescribed fundamental group

Abstract: In this article we study the problem of minimizing a? + bs on the class of all symplectic 4-manifolds with prescribed fundamental group G (? is the Euler characteristic, s is the signature, and a,?b ? R), focusing on the important cases ?, ? + s and 2? + 3s. In certain situations we can derive lower bounds for these functions and describe symplectic 4-manifolds which are minimizers. We derive an upper bound for the minimum of ? and ? + s in terms of the presentation of G.

The calabi-yau equation on almost-kähler four-manifolds

Abstract: Let (M, \omega) be a compact symplectic 4-manifold with a compatible almost complex structure J. The problem of finding a J-compatible symplectic form with prescribed volume form is an almost-K\"ahler analogue of Yau's theorem and is connected to a programme in symplectic topology proposed by Donaldson. We call the corresponding equation for the symplectic form the Calabi-Yau equation. Solutions are unique in their cohomology class. It is shown in this paper that a solution to this equation exists if the Nijenhuis tensor is small in a certain sense. Without this assumption, it is shown that the problem of existence can be reduced to obtaining a C^0 bound on a scalar potential function.

Arithmetic Quantum Unique Ergodicity for Symplectic Linear Maps of the

Abstract: We look at the expectation values for quantized linear symplectic maps on the multidimensional torus and their distribu- tion in the semiclassical limit. We construct super-scars that are stable under the arithmetic symmetries of the system and localize on invariant manifolds. We show that these super-scars exist only when there are isotropic rational subspaces, invariant under the linear map. In the case where there are no such scars, we com- pute the variance of the fluctuations of the matrix elements for the desymmetrized system, and present a conjecture for their limiting distributions.

Universal deformation formulae, symplectic lie groups and symmetric spaces

Abstract: We define a class of symplectic Lie groups associated with solvable symmetric spaces. We give a universal strict deformation formula for every proper action of such a group on a smooth manifold. We define a functional space where performing an asymptotic expansion of the nonformal deformed product in powers of the deformation parameter yields an associative formal star product on the symplectic Lie group at hand. The cochains of the star product are explicitly given ( without recursion) in the two-dimensional case of the affine group ax+b. The latter differs from the Giaquinto - Zhang construction, as shown by analyzing the invariance groups. In a Hopf algebra context, the above formal star product is shown to be a smash product and a compatible coproduct is constructed.

Subbundles of symplectic and orthogonal vector bundles over curves

Abstract: We review the notions of symplectic and orthogonal vector bundles over curves, and the connection between principal parts and extensions of vector bundles. We give a criterion for a certain extension of rank 2n to be symplectic or orthogonal. We then describe almost all of its rank n vector subbundles using graphs of sheaf homomorphisms, and give criteria for the isotropy of these subbundles. Finally, we sketch the use of these ideas in moduli questions for symplectic vector bundles. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Stratifications of Marsden-Weinstein reductions for representations of quivers and deformations of symplectic quotient singularities

Abstract: We investigate the Poisson geometry of the Marsden–Weinstein reductions of the moment map associated to the cotangent bundle of the space of representations of a quiver. We show that the stratification by representation type equals the stratification by symplectic leaves. The deformed symplectic quotient singularities—spectra of centres of symplectic reflection algebras associated to a wreath product—are shown to be isomorphic to reductions of a certain quiver. This establishes a method to calculate when these deformations are smooth. Furthermore, the isomorphism identifies symplectic leaves and so one can give a description of their symplectic leaves in terms of roots of the quiver.

Spectra of phase point operators in odd prime dimensions and the extended Clifford group

Abstract: We analyse the role of the Extended Clifford group in classifying the spectra of phase point operators within the framework laid out by Gibbons et al for setting up Wigner distributions on discrete phase spaces based on finite fields. To do so we regard the set of all the discrete phase spaces as a symplectic vector space over the finite field. Auxiliary results include a derivation of the conjugacy classes of ${\rm ESL}(2, \mathbb{F}_N)$.

Duality Through the Symplectic Embedding Formalism

Abstract: In this work we show that we can obtain dual equivalent actions following the symplectic formalism with the introduction of extra variables which enlarge the phase space. We show that the results are equal as the one obtained with the recently developed gauging iterative Noether dualization method. We believe that, with the arbitrariness property of the zero mode, the symplectic embedding method is more profound since it can reveal a whole family of dual equivalent actions. We illustrate the method demonstrating that the gauge-invariance of the electromagnetic Maxwell Lagrangian broken by the introduction of an explicit mass term and a topological term can be restored to obtain the dual equivalent and gauge-invariant version of the theory.