Rational SFT, Linearized Legendrian Contact Homology, and Lagrangian Floer Cohomology

Abstract

We relate the version of rational symplectic field theory for exact Lagrangian cobordisms introduced in [6] to linearized Legendrian contact homology. More precisely, if L ⊂ Xis an exact Lagrangian submanifold of an exact symplectic manifold with convex end Λ ⊂ Y, where Yis a contact manifold and Λis a Legendrian submanifold, and if Lhas empty concave end, then the linearized Legendrian contact cohomology of Λ, linearized with respect to the augmentation induced by L, equals the rational SFT of (X,L). Following ideas of Seidel [15], this equality in combination with a version of Lagrangian Floer cohomology of Lleads us to a conjectural exact sequence that in particular implies that if $X = {\mathbb{C}}^{n}$, then the linearized Legendrian contact cohomology of Λ ⊂ S^{2n − 1}is isomorphic to the singular homology of L. We outline a proof of the conjecture and show how to interpret the duality exact sequence for linearized contact homology of [7] in terms of the resulting isomorphism.