Symplectic Fibrations Research Articles

A field theory for symplectic fibrations over surfaces

We introduce in this paper a field theory on symplectic manifolds that are fibered over a real surface with interior marked points and cylindrical ends. We assign to each such object a morphism between certain tensor products of quantum and Floer homologies that are canonically attached to the fibration. We prove a composition theorem in the spirit of QFT, and show that this field theory applies naturally to the problem of minimising geodesics in Hofer's geometry. This work can be considered as a natural framework that incorporates both the Piunikhin-Salamon-Schwarz morphisms and the Seidel isomorphism.

Restrictions on symplectic fibrations

Moduli spaces of stable pseudoholomorphic curves can be defined parametrically, i.e., over total spaces of symplectic fibrations. This imposes several restrictions on the spectral sequence of a symplectic fibration. We prove, among others, that under certain assumptions the spectral sequence collapses at E 2. In the appendix, we prove nontriviality of certain Gromov–Witten invariant for blow-ups. As an application we obtain that any Hamiltonian fibration with the blow-up of CP 5 along four dimensional submanifold as a fibre c-splits. That is its spectral sequence collapses.

Homotopy decomposition of a group of symplectomorphisms of S2× S2

We continue the analysis started by Abreu, McDuff and Anjos of the topology of the group of symplectomorphisms of S 2× S 2 when the ratio of the area of the two spheres lies in the interval (1,2]. We express the group, up to homotopy, as the pushout (or amalgam) of certain of its compact Lie subgroups. We use this to compute the homotopy type of the classifying space of the group of symplectomorphisms and the corresponding ring of characteristic classes for symplectic fibrations.

Holomorphic Symplectic Manifolds and Lagrangian Fibrations

We introduce the geometrical nature of fibre space structures of an irreducible symplectic manifold and holomorphic Lagrangian fibrations.

Quantum Dynamical Yang-Baxter Equation¶Over a Nonabelian Base

In this paper we consider dynamical r-matrices over a nonabelian base. There are two main results. First, corresponding to a fat reductive decomposition of a Lie algebra ?=?⊕?, we construct geometrically a non-degenerate triangular dynamical r-matrix using symplectic fibrations. Second, we prove that a triangular dynamical r-matrix naturally corresponds to a Poisson manifold ?⋆×G. A special type of quantization of this Poisson manifold, called compatible star products in this paper, yields a generalized version of the quantum dynamical Yang–Baxter equation (or Gervais–Neveu–Felder equation). As a result, the quantization problem of a general dynamical r-matrix is proposed.

Loops of Lagrangian submanifolds and pseudoholomorphic discs

We examine three invariants of exact loops of Lagrangian submanifolds that are modelled on invariants introduced by Polterovich for loops of Hamiltonian symplectomorphisms. One of these is the minimal Hofer length in a given Hamiltonian isotopy class. We determine the exact values of these invariants for loops of projective Lagrangian planes. The proof uses the Gromov invariants of an associated symplectic fibration over the 2-disc with a Lagrangian subbundle over the boundary.

Gromov invariants of S2-bundles over 4-manifolds

We study the Gromov invariants of the total space W of a symplectic fibration π :W→M , where ( M, ω) is a symplectic 4-manifold and the fiber is equal to S 2. We find a relation between the Gromov invariants of W and those of M, for the homology classes A ̂ such that π( A ̂ )≠0 . As an application we construct infinitely many symplectic structures on W for M= E( n), the simply connected minimal elliptic surface.

D-branes, symplectomorphisms and noncommutative gauge theories

It is shown that the dual of the double compactified D=11 Supermembrane and a suitable compactified D=10 Super 4D-brane with nontrivial wrapping on the target space may be formulated as non-commutative gauge theories. The Poisson bracket over the world-volume is intrinsically defined in terms of the minima of the hamiltonian of the theory, which may be expressed in terms of a non degenerate 2-form. A deformation of the Poisson bracket in terms of the Moyal brackets is then performed. A non-commutative gauge theory in terms of the Moyal star bracket is obtained. It is shown that all these theories may be described in terms of symplectic connections on symplectic fibrations, the world volume being its base manifold and the (sub)group of volume preserving diffeomorphisms, p = 2 ( p > 2), generate the symplectomorphisms which preserve the (infinite dimensional) Poisson bracket of the fibration.

CompactifiedD=11supermembranes and symplectic noncommutative gauge theories

It is shown that a double compactified D=11 supermembrane with non trivial wrapping may be formulated as a symplectic non-commutative gauge theory on the world volume. The symplectic non commutative structure is intrinsically obtained from the symplectic 2-form on the world volume defined by the minimal configuration of its hamiltonian. The gauge transformations on the symplectic fibration are generated by the area preserving diffeomorphisms on the world volume. Geometrically, this gauge theory corresponds to a symplectic fibration over a compact Riemman surface with a symplectic connection.

A remark on almost complex quantization in symplectic fibrations

A remark on almost complex quantization in symplectic fibrations

Deformation Quantization of Symplectic Fibrations

A symplectic fibration is a fibre bundle in the symplectic category (a bundle of symplectic fibres over a symplectic base with a symplectic structure group). We find the relation between the deformation quantization of the base and the fibre, and that of the total space. We consider Fedosov's construction of deformation quantization. We generalize the Fedosov construction to the quantization with values in a bundle of algebras. We find that the characteristic class of deformation of a symplectic fibration is the weak coupling form of Guillemin, Lerman, and Sternberg. We also prove that the classical moment map could be quantized if there exists an equivariant connection.

The Geography Problem of 4-Manifolds with Various Structures

We review results about the geography problem of complex, symplectic and Einstein 4-manifolds. Finally we discuss the same problem for Lefschetz fibrations. We show that in the geography of symplectic 4-manifolds and Lefschetz fibrations the slopes α = c12/c2 do not admit gaps — following similar results of Sommese in the complex case.

Symplectic aspects of the first eigenvalue

There are two themes in the present paper. The first one is spelled out in the title, and is inspired by an attempt to find an analogue of Hersch-Yang-Yau estimate for $lambda_1$ of surfaces in symplectic category. In particular we prove that every split symplectic manifold $T^4 times M$ admits a compatible Riemannian metric whose first eigenvalue is arbitrary large. On the other hand for Kahler metrics compatible with a given integral symplectic form an upper bound for $lambda_1$ does exist. The second theme is the study of Hamiltonian symplectic fibrations over the 2-sphere. We construct a numerical invariant called the size of a fibration which arises as the solution of certain variational problems closely related to Hofer's geometry, K-area and coupling. In some examples it can be computed with the use of Gromov-Witten invariants. The link between these two themes is given by an observation that the first eigenvalue of a Riemannian metric compatible with a symplectic fibration admits a universal upper bound in terms of the size.

Sympletic normal connect sum

Since the publication in 1985 of Gromov’s paper [G1] on pseudo-holomorphic curves in symplectic manifolds there has been an increased interest in symplectic manifolds and symplectic topology. In particular, compact symplectic manifolds have become a focus of much study. In 1977 Thurston [T] gave an example of a compact symplectic manifold with first Betti number three, showing that not all compact symplectic manifolds admit a Kahler structure. However, the difference between the family of compact symplectic manifolds and compact Kahler manifolds remains unclear. In fact there are essentially only two general procedures for constructing compact symplectic manifolds: the symplectic fibration construction, originally due to Thurston, and blowing up along symplectic submanifolds, introduced by Gromov [G2]. Recently R. Gompf has introduced a new construction. He considers two symplectic 4-manifolds each containing the compact surface Σ symplectically embedded with trivial normal bundle. By the symplectic neighborhood theorem a tubular neighborhood of Σ in each 4-manifold is symplectomorphic to Σ×D2 equipped with the product symplectic structure. It follows then that the complements of the tubular neighborhoods of Σ in the symplectic 4-manifolds can be symplectically glued together along tubular shell neighborhoods of Σ by the map Id × φ where φ is an area preserving map of the annulus which interchanges the boundaries. Gompf proceeded by using this construction to show that a compact simply-connected 4-manifold not admitting any complex structure, which he constructed with T. Mrowka [GM], admits a symplectic structure. He thus produced the first example of a compact simply-connected symplectic 4-manifold not admitting any Kahler structure. In this paper we introduce a construction of four dimensional symplectic manifolds, that we call symplectic normal connect sum which generalizes Gompf’s construction. Our procedure constructs a new symplectic 4manifold X = X−1#ΨX1 from pairs (Xi,Σi), i = −1, 1, where the Xi are symplectic 4-manifolds and the Σi are compact embedded symplectic surfaces of genus g and of self-intersection n (for i = 1) and −n (for i = −1), n ≥ 0. We symplectically glue the complements of tubular neighborhoods of Σ−1 in X−1 and Σ1 in X1 along tubular shell neighborhoods of Σ−1 and

Fibered symplectic cohomology and the Leray-Serre spectral sequence

We define Symplectic cohomology groups for a class of symplectic fibrations with closed symplectic base and convex at infinity fiber. The crucial geometric assumption on the fibration is a negativity property reminiscent of negative curvature in complex vector bundles. When the base is symplectically aspherical we construct a spectral sequence of Leray-Serre type converging to the Symplectic cohomology groups of the total space, and we use it to prove new cases of the Weinstein conjecture.