Showing papers on "Symplectic manifold published in 2013"

Topological quantum field theory structure on symplectic cohomology

Abstract: We construct the TQFT on symplectic cohomology and wrapped Floer cohomology, possibly twisted by a local system of coefficients, and prove that the TQFT respects Viterbo restriction maps and the canonical maps from ordinary cohomology. We also construct the module structure of wrapped Floer cohomology over symplectic cohomology. These constructions yield new applications in symplectic topology relating to the Arnol'd chord conjecture and to exact contact embeddings. We prove that if a Liouville domain M admits an exact embedding into an exact convex symplectic manifold X, and the boundary of M is displaceable in X, then the symplectic cohomology of M vanishes and the chord conjecture holds for any Lagrangianly fillable Legendrian lying in the boundary. The TQFT respects the isomorphism between the symplectic cohomology of a cotangent bundle and the homology of the free loop space, so it recovers the TQFT of string topology. Finally, we use the TQFT to prove that symplectic cohomology vanishes iff Rabinowitz Floer cohomology vanishes.

Twisted cubics on cubic fourfolds

Abstract: We construct a new twenty-dimensional family of projective eight-dimensional irreducible holomorphic symplectic manifolds: the compactified moduli space M_3(Y) of twisted cubics on a smooth cubic fourfold Y that does not contain a plane is shown to be smooth and to admit a contraction M_3(Y) -> Z(Y) to a projective eight-dimensional symplectic manifold Z(Y). The construction is based on results on linear determinantal representations of singular cubic surfaces.

Deformation of the O'Grady moduli spaces

Abstract: In this paper we study moduli spaces of sheaves on an abelian or projective K3 surface. If S is a K3, v = 2w is a Mukai vector on S, where w is primitive and w(2) = 2, and H is a v-generic polarization on S, then the moduli space M-v of H-semistable sheaves on S whose Mukai vector is v admits a symplectic resolution (M) over tilde (v). A particular case is the 10-dimensional O'Grady example (M) over tilde (10) of an irreducible symplectic manifold. We show that (M) over tilde (v) is an irreducible symplectic manifold which is deformation equivalent to (M) over tilde (10) and that H-2 (M-v, Z) is Hodge isometric to the sublattice v(perpendicular to) of the Mukai lattice of S. Similar results are shown when S is an abelian surface.

Prime exceptional divisors on holomorphic symplectic varieties and monodromy reflections

Abstract: Let X be a projective irreducible holomorphic symplectic manifold. The second integral cohomology of X is a lattice with respect to the Beauville–Bogomolov pairing. A divisor E on X is called a prime exceptional divisor if E is reduced and irreducible and of negative Beauville–Bogomolov degree. Let E be a prime exceptional divisor on X. We first observe that associated to E is a monodromy involution of the integral cohomology H∗(X,Z), which acts on the second cohomology lattice as the reflection by the cohomology class [E] of E. We then specialize to the case where X is deformation equivalent to the Hilbert scheme of length n zero-dimensional subschemes of a K3 surface, n≥2. We determine the set of classes of exceptional divisors on X. This leads to a determination of the closure of the movable cone of X.

The Gopakumar-Vafa formula for symplectic manifolds

Abstract: The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic Gromov-Witten theory, we prove that the Gopakumar-Vafa formula holds for any symplectic Calabi-Yau 6-manifold, and hence for Calabi-Yau 3-folds. The results extend to all symplectic 6-manifolds and to the genus zero GW invariants of semipositive manifolds.

2-plectic geometry, Courant algebroids, and categorified prequantization

Abstract: A 2-plectic manifold is a manifold equipped with a closed nondegenerate 3-form, just as a symplectic manifold is equipped with a closed nondegenerate 2-form. In 2-plectic geometry one finds the higher analogues of many structures familiar from symplectic geometry. For example, any 2-plectic manifold has a Lie 2-algebra consisting of smooth functions and Hamiltonian 1-forms. This is equipped with a Poisson-like bracket which only satisfies the Jacobi identity up to “coherent chain homotopy”. Over any 2-plectic manifold is a vector bundle equipped with extra structure called an exact Courant algebroid. This Courant algebroid is the 2-plectic analogue of a transitive Lie algebroid over a symplectic manifold. Its space of global sections also forms a Lie 2-algebra. We show that this Lie 2-algebra contains an important sub-Lie 2-algebra which is isomorphic to the Lie 2-algebra of Hamiltonian 1-forms. Furthermore, we prove that it is quasi-isomorphic to a central extension of the (trivial) Lie 2-algebra of Hamiltonian vector fields, and therefore is the higher analogue of the well-known Kostant–Souriau central extension in symplectic geometry. We interpret all of these results within the context of a categorified prequantization procedure for 2-plectic manifolds. In doing so, we describe how $U(1)$-gerbes, equipped with a connection and curving, and Courant algebroids are the 2-plectic analogues of principal $U(1)$ bundles equipped with a connection and their associated Atiyah Lie algebroids.

Uniqueness of symplectic structures

Abstract: This survey paper discusses some uniqueness questions for symplectic forms on compact manifolds without boundary.

Moduli of K3 Surfaces and Irreducible Symplectic Manifolds

Abstract: The name "K3 surfaces" was coined by A. Weil in 1957 when he formulated a research programme for these surfaces and their moduli. Since then, irreducible holomorphic symplectic manifolds have been introduced as a higher dimensional analogue of K3 surfaces. In this paper we present a review of this theory starting from the definition of K3 surfaces and going as far as the global Torelli theorem for irreducible holomorphic symplectic manifolds as recently proved by M. Verbitsky. For many years the last open question of Weil's programme was that of the geometric type of the moduli spaces of polarised K3 surfaces. We explain how this problem has been solved. Our method uses algebraic geometry, modular forms and Borcherds automorphic products. We collect and discuss the relevant facts from the theory of modular forms with respect to the orthogonal group O(2,n). We also give a detailed description of quasi pull-back of automorphic Borcherds products. This part contains previously unpublished results. We apply our geometric-automorphic method to study moduli spaces of both polarised K3 surfaces and irreducible symplectic varieties.

Hofer’s metrics and boundary depth

Abstract: We show that if (M ,ω) is a closed symplectic manifold which admits a nontrivial Hamiltonian vector field all of whose contractible closed orbits are constant, then Hofer’s metric on the group of Hamiltonian diffeomorphisms of (M ,ω) has infinite diameter, and indeed admits infinite-dimensional quasi-isometrically embedded normed vector spaces. A similar conclusion applies to Hofer’s metric on various spaces of Lagrangian submanifolds, including those Hamiltonian-isotopic to the diagonal in M × M when M satisfies the above dynamical condition. To prove this, we use the properties of a Floer-theoretic quantity called the boundary depth, which measures the nontriviality of the boundary operator on the Floer complex in a way that encodes robust symplectic-topological information.

Open Gromov-Witten disk invariants in the presence of an anti-symplectic involution

Abstract: For a symplectic manifold with an anti-symplectic involution having non-empty fixed locus, we construct a model of the moduli space of real sphere maps out of moduli spaces of decorated disk maps and give an explicit expression for its first Stiefel-Whitney class. As a corollary, we obtain a large number of examples, which include all odd-dimensional projective spaces and many complete intersections, for which many types of real moduli spaces are orientable. For these manifolds, we define open Gromov-Witten invariants with no restriction on the dimension of the manifolds or the type of the constraints if there are no boundary marked points. If there are boundary marked points, we define the invariants under some restrictions on the allowed boundary constraints, even though the moduli spaces are not orientable in these cases.

On non‐contractible periodic orbits of Hamiltonian diffeomorphisms

Abstract: We prove that any Hamiltonian diffeomorphism of a closed symplectic manifold equipped with an atoroidal symplectic form has simple non-contractible periodic orbits of arbitrarily large period, provided that the diffeomorphism has a non-degenerate (or even isolated and homologically non-trivial) periodic orbit with non-zero homology class and the set of one-periodic orbits in that class is finite.

Local classification of generalize complex structures

Abstract: We give a local classification of generalized complex structures. About a point, a generalized complex structure is equivalent to a product of a symplectic manifold with a holomorphic Poisson manifold. We use a Nash-Moser type argument in the style of Conn’s linearization theorem.

On cohomological obstructions to the existence of log symplectic structures

Abstract: We prove that a compact log symplectic manifold has a class in the second cohomology group whose powers, except maybe for the top, are nontrivial This result gives cohomological obstructions for the existence of b-log symplectic structures similar to those in symplectic geometry

Virtual neighborhood technique for pseudo-holomorphic spheres

Abstract: This is the first part of a trilogy where we apply the theory of virtual manifold/orbifolds developed by the first named author and Tian to study the Gromov-Witten moduli spaces. In this paper, we resolve the main analytic issue arising from the lack of differentiability of $PSL(2, \C)$-action on spaces of $W^{1, p}$-maps from the Riemann sphere to a symplectic manifold $(X, \omega, J)$ with a non-zero homology class $A$. In particular, we establish the slice and tubular neighbourhood theorems for $PSL(2, \C)$-action along smooth maps, and construct a $PSL(2, \C)$-obstruction bundle along $PSL(2, \C)$-orbit of a pseudo-holomorphic map representing a point in the moduli space $\cM_{0, 0}(X, A)$. In Sections 2 and 3 of this paper, we explain an integration theory on virtual orbifolds using proper \'etale groupoids and establish the virtual neighborhood technique for a general orbifold Fredholm system. When the moduli space $\cM_{0, 0}(X, A)$ of pseudo-holomorphic spheres in $(X, \omega, J)$ is compact, applying the virtual neighborhood technique developed in Section 3, we obtain a virtual system for the moduli space $\cM_{0, 0}(X, A)$ of pseudo-holomorphic spheres in $(X, \omega, J)$ and show that the genus zero Gromov-Witten invariant is well-defined.

Relational symplectic groupoids and Poisson sigma models with boundary

Abstract: We introduce the notion of relational symplectic groupoid as a way to integrate Poisson manifolds in general, following the construction through the Poisson sigma model (PSM) given by Cattaneo and Felder. We extend such construction to the infinite dimensional setting, where we are able to describe the relational symplectic groupoid in terms of immersed Lagrangian Bannach submanifolds. This corresponds to a groupoid object in the "extended symplectic category" and it can be related to the ususal version of symplectic groupoids via reduction. We prove the existence of such an object for any Poisson manifold and the uniqueness of a compatible Poisson structure on the base, for a special type of relational symplectic groupoids. We develop the notion of equivalence, that allows us to compare finite and infinite dimensional examples. We discuss some other extensions to different categories, where the relational construction still make sense, in an effort to understand better geometric quantization of symplectic manifolds in this new perspective.

Lifting pseudo-holomorphic polygons to the symplectisation of $P \times \mathbb{R}$ and applications

Abstract: Let $\mathbb{R} \times (P \times \mathbb{R})$ be the symplectisation of the contactisation of an exact symplectic manifold $P$, and let $\mathbb{R} \times \Lambda$ be a cylinder over a Legendrian submanifold in the contactisation We show that a pseudo-holomorphic polygon in $P$ having boundary on the projection of $\Lambda$ can be lifted to a pseudo-holomorphic disc in the symplectisation having boundary on $\mathbb{R} \times \Lambda$ It follows that Legendrian contact homology may be equivalently defined by counting either of these objects Using this result, we give a proof of Seidel's isomorphism of the linearised Legendrian contact homology induced by an exact Lagrangian filling and the singular homology of the filling

Quantum mechanics with coordinate dependent noncommutativity

Abstract: Noncommutative quantum mechanics can be considered as a first step in the construction of quantum field theory on noncommutative spaces of generic form, when the commutator between coordinates is a function of these coordinates. In this paper we discuss the mathematical framework of such a theory. The noncommutativity is treated as an external antisymmetric field satisfying the Jacobi identity. First, we propose a symplectic realization of a given Poisson manifold and construct the Darboux coordinates on the obtained symplectic manifold. Then we define the star product on a Poisson manifold and obtain the expression for the trace functional. The above ingredients are used to formulate a nonrelativistic quantum mechanics on noncommutative spaces of general form. All considered constructions are obtained as a formal series in the parameter of noncommutativity. In particular, the complete algebra of commutation relations between coordinates and conjugated momenta is a deformation of the standard Heisenberg algebra. As examples we consider a free particle and an isotropic harmonic oscillator on the rotational invariant noncommutative space.

Quasi-states, quasi-morphisms, and the moment map

Abstract: We prove that symplectic quasi-states and quasi-morphisms on a symplectic manifold descend under symplectic reduction on a superheavy level set of a Hamiltonian torus action. Using a construction due to Abreu and Macarini, in each dimension at least four we produce a closed symplectic toric manifold with infinite dimensional spaces of symplectic quasi-states and quasi-morphisms, and a one-parameter family of non-displaceable Lagrangian tori. By using McDuff's method of probes, we also show how Ostrover and Tyomkin's method for finding distinct spectral quasi-states in symplectic toric Fano manifolds can also be used to find different superheavy toric fibers.

The diversity of symplectic calabi-yau 6-manifolds

Abstract: Given an integer b and a finitely presented group G, we produce a compact symplectic 6-manifold with c1 = 0, b2 > b, b3 > b and pi = G. In the simply connected case, we can also arrange for b3 = 0; in particular, these examples are not diffeomorphic to Kahler manifolds with c1 = 0. The construction begins with a certain orientable, four-dimensional, hyperbolic orbifold assembled from right-angled 120-cells. The twistor space of the hyperbolic orbifold is a symplectic Calabi- Yau orbifold; a crepant resolution of this last orbifold produces a smooth symplectic manifold with the required properties. © 2013 London Mathematical Society.

Lagrangian floer Theory over integers: spherically positive symplectic manifolds

Abstract: In this paper we study the Lagrangian Floer theory over Z or Z2. Under an appropriate assumption on ambient symplectic manifold, we show that the whole story of Lagrangian Floer theory in [6], [7] can be developed over Z2 coefficients, and over Z coefficients when Lagrangian submanifolds are relatively spin. The main technical tools used for the construction are the notion of the sheaf of groups, and stratification and compatibility of the normal cones applied to the Kuranishi structure of the moduli space of pseudo-holomorphic discs.

Symplectic Bott-Chern cohomology of solvmanifolds

Abstract: We study the symplectic Bott-Chern cohomology by L.-S. Tseng and S.-T. Yau for solvmanifolds endowed with left-invariant symplectic structures. Our results are applicable to cohomology with values in local systems. Studying symplectic Bott-Chern cohomology of solvmanifolds with values in local systems, we give some remarks on symplectic Hodge theory.

The orientability problem in open Gromov–Witten theory

Abstract: We give an explicit formula for the holonomy of the orientation bundle of a family of real Cauchy‐Riemann operators. A special case of this formula resolves the orientability question for spaces of maps from Riemann surfaces with Lagrangian boundary condition. As a corollary, we show that the local system of orientations on the moduli space of J‐holomorphic maps from a bordered Riemann surface to a symplectic manifold is isomorphic to the pullback of a local system defined on the product of the Lagrangian and its free loop space. As another corollary, we show that certain natural bundles over these moduli spaces have the same local systems of orientations as the moduli spaces themselves (this is a prerequisite for integrating the Euler classes of these bundles). We will apply these conclusions in future papers to construct and compute open Gromov‐Witten invariants in a number of settings. 53D45, 57R17; 14N35

On isotropic divisors on irreducible symplectic manifolds

Abstract: Let X be an irreducible symplectic manifold and L a divisor on X. Assume that L is isotropic with respect to the Beauville-Bogomolov quadratic form. We define the rational Lagrangian locus and the movable locus on the universal deformation space of the pair (X, L). We prove that the rational Lagrangian locus is empty or coincide with the movable locus.

Dualities near the horizon

Abstract: In 4-dimensional supergravity theories, covariant under symplectic electricmagnetic duality rotations, a significant role is played by the symplectic matrix $ \mathcal{M} $ (φ), related to the coupling of scalars φ to vector field-strengths. In particular, this matrix enters the twisted self-duality condition for 2-form field strengths in the symplectic formulation of generalized Maxwell equations in the presence of scalar fields. In this investigation, we compute several properties of this matrix in relation to the attractor mechanism of extremal (asymptotically flat) black holes. At the attractor points with no flat directions (as in the $ \mathcal{N} $ = 2 BPS case), this matrix enjoys a universal form in terms of the dyonic charge vector $ \mathcal{Q} $ and the invariants of the corresponding symplectic representation $ {R_{\mathcal{Q}}} $ of the duality group G, whenever the scalar manifold is a symmetric space with G simple and non-degenerate of type E7. At attractors with flat directions, $ \mathcal{M} $ still depends on flat directions, but not $ \mathcal{M}\mathcal{Q} $ , defining the so-called Freudenthal dual of $ \mathcal{Q} $ itself. This allows for a universal expression of the symplectic vector field strengths in terms of $ \mathcal{Q} $ , in the near-horizon Bertotti-Robinson black hole geometry.

Semi-classical properties of Berezin--Toeplitz operators with $\cC^k$--\,symbol

Abstract: We obtain the semi-classical expansion of the kernels and traces of Toeplitz operators with $\cC^k$--\,symbol on a symplectic manifold. We also give a semi-classical estimate of the distance of a Toeplitz operator to the space of self-adjoint and multiplication operators.

Symplectic half-flat solvmanifolds

Abstract: We classify solvable Lie groups admitting left invariant symplectic half-flat structure. When the Lie group has a compact quotient by a lattice, we show that these structures provide solutions of supersymmetric equations of type IIA.

Relativistic Kinetic Theory: An Introduction

Abstract: We present a brief introduction to the relativistic kinetic theory of gases with emphasis on the underlying geometric and Hamiltonian structure of the theory. Our formalism starts with a discussion on the tangent bundle of a Lorentzian manifold of arbitrary dimension. Next, we introduce the Poincare one-form on this bundle, from which the symplectic form and a volume form are constructed. Then, we define an appropriate Hamiltonian on the bundle which, together with the symplectic form yields the Liouville vector field. The corresponding flow, when projected onto the base manifold, generates geodesic motion. Whenever the flow is restricted to energy surfaces corresponding to a negative value of the Hamiltonian, its projection describes a family of future-directed timelike geodesics. A collisionless gas is described by a distribution function on such an energy surface, satisfying the Liouville equation. Fibre integrals of the distribution function determine the particle current density and the stress-energy tensor. We show that the stress-energy tensor satisfies the familiar energy conditions and that both the current and stress-energy tensor are divergence-free. Our discussion also includes the generalization to charged gases, a summary of the Einstein-Maxwell-Vlasov system in any dimensions, as well as a brief introduction to the general relativistic Boltzmann equation for a simple gas.

Action–index relations for perfect Hamiltonian diffeomorphisms

Abstract: We show that the actions and indexes of fixed points of a Hamiltonian diffeomorphism with finitely many periodic points must satisfy certain relations, provided that the quantum cohomology of the ambient manifold meets an algebraic requirement satisfied for projective spaces, Grassmannians and many other manifolds. We also refine a previous result on the Conley conjecture for negative monotone symplectic manifolds, due to the second and third authors, and show that a Hamiltonian diffeomorphism of such a manifold must have simple periodic orbits of arbitrarily large period whenever its fixed points are isolated.