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

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 $\mathbb{Z}$-graded chain complex of vector spaces over ${\mathbb{Z}}_2$, 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 $(X^{\prime },L^{\prime })$ to a positive end of a pair $(X^{\prime \prime },L^{\prime \prime })$, then there are natural morphisms from the spectral sequences of $(X^{\prime },L^{\prime })$ and of $(X^{\prime \prime },L^{\prime \prime })$ 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 $(Y\times \mathbb{R},{\Lambda }_k^s\times \mathbb{R})$, where $Y\times \mathbb{R}$ 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 $\Lambda$.