Rational Symplectic Field Theory over Z2 for Exact Lagrangian Cobordisms

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,Λ k ×R), where Y ×R is the symplectization of Y and where Λ k ⊂ Y is the Legendrian submanifold consisting of s parallel copies of Λ subdivided into k subsets, give Legendrian isotopy invariants of Λ.