Coherent orientations in symplectic field theory

Abstract. A symplectic manifold W with contact type boundary M = ∂W induces a linearization of the contact homology of M with corresponding linearized contact homology HC(M ). We establish a Gysin-type exact sequence in which the symplectic homology SH(W ) of W maps to HC(M ), which in turn maps to HC(M ), by a map of degree −2, which then maps to SH(W ). Furthermore, we give a description of the degree −2 map in terms of rational holomorphic curves with constrained asymptotic markers, in the symplectization of M . ContentsSuch an X is called a Liouville vector field. The 1-form λ := (ι X ω)| M is a contact form on M . We denote by ξ the contact distribution defined by λ and we call (W, ω) a filling of (M, ξ).We assume throughout the paper that (W, ω) satisfies the conditionT 2 f * ω = 0 for all smooth f :where T 2 is the 2-torus. This condition guarantees that the energy of a Floer trajectory (for a definition, see for example [7, Section 2]) does not depend on its homology class, but only on its endpoints. Our main class of examples is provided by exact symplectic forms.Theorem 1 ties together the symplectic homology groups of (W, ω) and the linearized contact homology groups of (M, ξ). Both these invariants encode algebraically the dynamics of the same vector field, the Reeb vector field R λ defined by ker ω| M = R λ and λ(R λ ) = 1. But their natures are quite different: the former belongs to the realm of Floer theory [17,32], whereas the latter belongs to the realm of symplectic field theory (SFT) [16]. Our result can be read as a way to make symplectic homology fit into SFT.Let us introduce some relevant notation. Given a free homotopy class a of loops in W we denote by SH a * (W, ω) the symplectic homology groups of (W, ω) in the homotopy class a. The free homotopy class of the constant loop will be denoted by 0. We also denote by SH + * (W, ω) the symplectic homology groups in the trivial homotopy class truncated at a small positive value of the action functional. We refer to Section 2 for the definitions.Let i : M ֒→ W be the inclusion. Given a free homotopy class a of loops in W we denote by i −1 (a) the set of free homotopy classes in M which are mapped to a via i, and we use the convention i −1 (+) := i −1 (0). We denote by HC i −1 (a) * (M, ξ) the linearized contact homology groups of (M, ξ) based on closed Reeb orbits whose free homotopy class belongs to i −1 (a). We refer to Section 3.1 for the definition.Both SH a * (W, ω) and HC i −1 (a) * (M, ξ) are defined over the Novikov ring Λ ω with Q-coefficients consisting of formal combinations λ := A∈H2(W ;Z) λ A e A , λ A ∈ Q such that #{A|λ A = 0, ω(A) ≤ c} < ∞ for all c > 0. The multiplication in Λ ω is given by the convolution product.We assume the existence of an almost complex structure J such that linearized contact homology is defined. This means that J needs to be regular for rigid holomorphic planes in the symplectic completion of W , as well as for rational holomorphic curves with one positive puncture in the symplectization of M satisfyin...

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“…One can associate a sign ǫ(F ) to each element [F ] of this moduli space [6] and we define the linearized contact differential…”

Section: One Can Associate a Sign ǫ(F ) To Each Element [F ] ∈ Mmentioning

confidence: 99%

An exact sequence for contact- and symplectic homology

2008

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“…Each of these m cylinders sits in R '.S 1 D/ as the image of the graph over R S 1 of the function that sends .s; t/ section of this bundle, thus a C -valued function on C . This pair obeys the equations (1)(2)(3)(4)(5) F A D i .1 j˛j 2 /.…”

Section: A Relative Gradings and The Mapˆrmentioning

confidence: 99%

“…It also comes with a natural, complete Kahler metric; but this is not the flat Kahler metric unless m D 1. In any event, this metric defines a symplectic form and thus the Hamiltonian dynamical system that is defined using the time dependent Hamiltonian function (1)(2)(3)(4)(5)(6) h D The spectrum of this operator is a discrete subset of R with finite multiplicities and no accumulation points. What is denoted here by C‚ consists of the elements in C‚ of the form fc g .…”

Section: A Relative Gradings and The Mapˆrmentioning

confidence: 99%

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Embedded contact homology and Seiberg–Witten Floer cohomology III

Taubes

2010

Geom. Topol.

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“…In particular, they showed that the fundamental group π 1 (Ξ(T 3 ), ξ n ) based at any tight contact structure ξ n on T 3 contains an infinite cyclic group. They obtain a similar result in the more general case of T 2 -bundles over S 1 .…”

Section: Introductionmentioning

confidence: 99%