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

Source abstract

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 .