Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology

TL;DR

This paper establishes a deep connection between rational Symplectic Field Theory, linearized Legendrian contact homology, and Lagrangian Floer cohomology, providing new insights into their relationships and conjectural exact sequences in symplectic topology.

Contribution

It relates rational SFT for exact Lagrangian cobordisms to linearized Legendrian contact homology and proposes a conjectural exact sequence linking these theories.

Findings

Linearized Legendrian contact cohomology equals rational SFT for certain Lagrangian submanifolds.

Conjectural exact sequence connecting Floer cohomology and contact homology.

Isomorphism between Legendrian contact cohomology and singular homology in specific cases.

Abstract

We relate the version of rational Symplectic Field Theory for exact Lagrangian cobordisms introduced in [5] with linearized Legendrian contact homology. More precisely, if $L\subset X$ is an exact Lagrangian submanifold of an exact symplectic manifold with convex end $\Lambda\subset Y$, where $Y$ is a contact manifold and $\Lambda$ is a Legendrian submanifold, and if $L$ has empty concave end, then the linearized Legendrian contact cohomology of $\Lambda$, linearized with respect to the augmentation induced by $L$, equals the rational SFT of $(X,L)$. Following ideas of P. Seidel, this equality in combination with a version of Lagrangian Floer cohomology of $L$ leads us to a conjectural exact sequence which in particular implies that if $X=\C^{n}$ then the linearized Legendrian contact cohomology of $\Lambda\subset S^{2n-1}$ 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 [6] in terms of the resulting isomorphism.