# Rational symplectic field theory over $Z_{2}$ for exact Lagrangian cobordisms

### Tobias Ekholm

Uppsala Universitet, Sweden

## Abstract

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 $Z_{2}$, 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 $Λ⊂Y$ 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 $Λ$.

## Cite this article

Tobias Ekholm, Rational symplectic field theory over $Z_{2}$ for exact Lagrangian cobordisms. J. Eur. Math. Soc. 10 (2008), no. 3, pp. 641–704

DOI 10.4171/JEMS/126