Rational symplectic field theory over $\mathbb{Z}_2$ for exact Lagrangian cobordisms

Tobias Ekholm

Journal of the European Mathematical Society, Tome 10 (2008) no. 3, pp. 641-704.

Résumé

We construct a version of rational Symplectic Field Theory for pairs (X,L), where X is an exact symplectic manifold, where L⊂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 Z2​, 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 Λ⊂Y is a Legendrian submanifold of a contact manifold then the spectral sequences associated to (Y×R,Λks​×R), where Y×R is the symplectization of Y and where Λks​⊂Y is the Legendrian submanifold consisting of s parallel copies of Λ subdivided into k subsets, give Legendrian isotopy invariants of Λ.

Détail

DOI : 10.4171/jems/126

Classification : 57-XX, 53-XX, 00-XX
Keywords: Holomorphic curve, Lagrangian submanifold, Legendrian submanifold, symplectic cobordism, symplectic field theory