165 research outputs found

    A cocycle on the group of symplectic diffeomorphisms

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    We define a cocycle on the group of symplectic diffeomorphisms of a symplectic manifold and investigate its properties. The main applications are concerned with symplectic actions of discrete groups. For example, we give an alternative proof of the Polterovich theorem about the distortion of cyclic subgroups in finitely generated groups of Hamiltonian diffeomorphisms.Comment: 19 pages, no figures, corrected versio

    Discontinuous symplectic capacities

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    We show that the spherical capacity is discontinuous on a smooth family of ellipsoidal shells. Moreover, we prove that the shell capacity is discontinuous on a family of open sets with smooth connected boundaries.Comment: We include generalizations to higher dimensions due to the unknown referee and Janko Latschev. We add examples of open sets with connected boundary on which the shell capacity is not continuous. 3rd and 4th version: minor changes, to appear in J. Fixed Point Theory App

    Massey products in symplectic manifolds

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    The paper is devoted to study of Massey products in symplectic manifolds. Theory of generalized and classical Massey products and a general construction of symplectic manifolds with nontrivial Massey products of arbitrary large order are exposed. The construction uses the symplectic blow-up and is based on the author results, which describe conditions under which the blow-up of a symplectic manifold X along its submanifold Y inherits nontrivial Massey products from X ot Y. This gives a general construction of nonformal symplectic manifolds.Comment: LaTeX, 48 pages, 2 figure

    Birational cobordism invariance of uniruled symplectic manifolds

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    A symplectic manifold (M,ω)(M,\omega) is called {\em (symplectically) uniruled} if there is a nonzero genus zero GW invariant involving a point constraint. We prove that symplectic uniruledness is invariant under symplectic blow-up and blow-down. This theorem follows from a general Relative/Absolute correspondence for a symplectic manifold together with a symplectic submanifold. A direct consequence is that symplectic uniruledness is a symplectic birational invariant. Here we use Guillemin and Sternberg's notion of cobordism as the symplectic analogue of the birational equivalence.Comment: To appear in Invent. Mat

    New obstructions to symplectic embeddings

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    In this paper we establish new restrictions on symplectic embeddings of certain convex domains into symplectic vector spaces. These restrictions are stronger than those implied by the Ekeland-Hofer capacities. By refining an embedding technique due to Guth, we also show that they are sharp.Comment: 80 pages, 3 figures, v2: improved exposition and minor corrections, v3: Final version, expanded and improved exposition and minor corrections. The final publication is available at link.springer.co

    On the structure of Clifford quantum cellular automata

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    We study reversible quantum cellular automata with the restriction that these are also Clifford operations. This means that tensor products of Pauli operators (or discrete Weyl operators) are mapped to tensor products of Pauli operators. Therefore Clifford quantum cellular automata are induced by symplectic cellular automata in phase space. We characterize these symplectic cellular automata and find that all possible local rules must be, up to some global shift, reflection invariant with respect to the origin. In the one dimensional case we also find that every uniquely determined and translationally invariant stabilizer state can be prepared from a product state by a single Clifford cellular automaton timestep, thereby characterizing these class of stabilizer states, and we show that all 1D Clifford quantum cellular automata are generated by a few elementary operations. We also show that the correspondence between translationally invariant stabilizer states and translationally invariant Clifford operations holds for periodic boundary conditions.Comment: 28 pages, 2 figures, LaTe

    Topological Field Theory and Rational Curves

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    We analyze the superstring propagating on a Calabi-Yau threefold. This theory naturally leads to the consideration of Witten's topological non-linear sigma-model and the structure of rational curves on the Calabi-Yau manifold. We study in detail the case of the world-sheet of the string being mapped to a multiple cover of an isolated rational curve and we show that a natural compactification of the moduli space of such a multiple cover leads to a formula in agreement with a conjecture by Candelas, de la Ossa, Green and Parkes.Comment: 20 page

    Mirror Manifolds in Higher Dimension

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    We describe mirror manifolds in dimensions different from the familiar case of complex threefolds. We emphasize the simplifying features of dimension three and supply more robust methods that do not rely on such special characteristics and hence naturally generalize to other dimensions. The moduli spaces for Calabi--Yau dd-folds are somewhat different from the ``special K\"ahler manifolds'' which had occurred for d=3d=3, and we indicate the new geometrical structures which arise. We formulate and apply procedures which allow for the construction of mirror maps and the calculation of order-by-order instanton corrections to Yukawa couplings. Mathematically, these corrections are expected to correspond to calculating Chern classes of various parameter spaces (Hilbert schemes) for rational curves on Calabi--Yau manifolds. Our results agree with those obtained by more traditional mathematical methods in the limited number of cases for which the latter analysis can be carried out. Finally, we make explicit some striking relations between instanton corrections for various Yukawa couplings, derived from the associativity of the operator product algebra.Comment: 44 pages plus 3 tables using harvma

    Gauging and symplectic blowing up in nonlinear sigma-models: I. point singularities

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    In this paper a two dimensional non-linear sigma model with a general symplectic manifold with isometry as target space is used to study symplectic blowing up of a point singularity on the zero level set of the moment map associated with a quasi-free Hamiltonian action. We discuss in general the relation between symplectic reduction and gauging of the symplectic isometries of the sigma model action. In the case of singular reduction, gauging has the same effect as blowing up the singular point by a small amount. Using the exponential mapping of the underlying metric, we are able to construct symplectic diffeomorphisms needed to glue the blow-up to the global reduced space which is regular, thus providing a transition from one symplectic sigma model to another one free of singularities.Comment: 32 pages, LaTex, THEP 93/24 (corrected and expanded(about 5 pages) version

    Supersymmetric Wilson loops on S^3

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    This paper studies in great detail a family of supersymmetric Wilson loop operators in N=4 supersymmetric Yang-Mills theory we have recently found. For a generic curve on an S^3 in space-time the loops preserve two supercharges but we will also study special cases which preserve 4, 8 and 16 supercharges. For certain loops we find the string theory dual explicitly and for the general case we show that string solutions satisfy a first order differential equation. This equation expresses the fact that the strings are pseudo-holomorphic with respect to a novel almost complex structure we construct on AdS_4 x S^2. We then discuss loops restricted to S^2 and provide evidence that they can be calculated in terms of similar observables in purely bosonic YM in two dimensions on the sphere.Comment: Latex, 84 pages, 4 figures. v2: minor changes, references added; to appear in JHE
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