Rational symplectic field theory over Z2 for exact Lagrangian cobordisms

By T. Ekholm. 2006

82 citations recorded by the source corpus.

We construct a version of rational Symplectic Field Theory for pairs $(X,L)$, where $X$ is an exact symplectic manifold, where $L\subset X$ is an exact Lagrangian submanifold with components subdivided into $k$ subsets, and where both $X$ and $L$ have cylindrical ends. The theory associates to $(X,L)$ a $\Z$-graded chain complex of vector spaces over $\Z_2$, filtered with $k$ filtration levels. The corresponding $k$-level spectral sequence is invariant under deformations of $(X,L)$ and has the following property: if $(X,L)$ is obtained by joining a negative end of a pair $(X',L')$ to a positive end of a pair $(X'',L'')$, then there are natural morphisms from the spectral sequences of $(X',L')$ and of $(X'',L'')$ to the spectral sequence of $(X,L)$. As an application, we show that if $\Lambda\subset Y$ is a Legendrian submanifold of a contact manifold then the spectral sequences associated to $(Y\times\R,\Lambda_k^s\times\R)$, where $Y\times\R$ is the symplectization of $Y$ and where $\Lambda_k^s\subset Y$ is the Legendrian submanifold consisting of $s$ parallel copies of $\Lambda$ subdivided into $k$ subsets, give Legendrian isotopy invariants of $\Lambda$.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b4308f1be7f6e9bf412ba6ee6be446f03ca2ee54

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Citations in the corpus

Duality between Lagrangian and Legendrian invariants (2017, background): “, [28, 23, 11].”
Lagrangian cobordisms via generating families: Construction and geography (2014, unknown): “…their centrality in symplectic ﬁeld theory (see Eliashberg, Givental and Hofer [19], and see also Bourgeois, Ekholm and Eliashberg [7] and Ekholm [13; 14] for recent examples) and wrapped Fukaya category frameworks (see Abbondandolo and Schwarz [1], Abouzaid and Seidel [2], and Fukaya, Seidel…”
Obstructions to Lagrangian concordance (2014, background): “In symplectic and contact topology, there has been a great deal of recent interest in the subject of Lagrangian cobordisms between Legendrian submanifolds; see for example [1, 3, 5, 6, 8, 9, 13, 14, 26, 27, 44].”
Effect of Legendrian surgery (2009, background): “The authors acknowledge the support of: F. Bourgeois:The Fonds National de la Recherche Scientifique (Belgium) T. Ekholm: The G̈oran Gustafsson Foundation for Research in Natural Sciences and Medicine Y. Eliashberg:NSF grants DMS-0707103 and DMS 0244663”
Rational SFT, Linearized Legendrian Contact Homology, and Lagrangian Floer Cohomology (2009, methodology): “We relate the version of rational Symplectic Field Theory for exact Lagrangian cobordisms introduced in [5] with linearized Legendrian contact homology.”
Legendrian ambient surgery and Legendrian contact homology (2012, methodology): “It follows by the results in [Ekh] that the homotopy class of ΦV is invariant under compactly supported deformations of the almost complex structure and compactly supported Hamiltonian isotopies of V (see Theorem 3.”
Altering symplectic manifolds by homologous recombination (2010, background): “This is not really surprising (compare for instance [17], which introduced rational Symplectic Field Theory for Lagrangian submanifolds with coefficients in K = F2), and only requires careful inspection of the moduli spaces involved.”