$A_\infty$ Sabloff Duality via the LSFT Algebra

TL;DR

This paper advances the understanding of Legendrian knot invariants by establishing an $A_ abla$-algebraic duality and exact sequences within the positive augmentation category, using the LSFT algebra framework.

Contribution

It introduces an $A_ abla$-algebraic Sabloff duality, extends exact sequences to bimodules, and defines curved augmentations to construct homotopy inverses and higher homotopies.

Findings

Establishes an $A_ abla$-algebraic Sabloff duality as a quasi-isomorphism.

Extends the Ekholm-Etnyre-Sabloff exact sequence to bimodules.

Defines curved augmentations to construct homotopy inverses and higher homotopies.

Abstract

We use Ng's LSFT algebra to upgrade Sabloff duality of Legendrian knots to a quasi-isomorphism of $A_\infty$ bimodules over the positive augmentation category $\mathcal{A}ug_+$. We also extend the Ekholm-Etnyre-Sabloff exact sequence to an exact sequence of $\mathcal{A}ug_+$-bimodules, using a quotient category $\mathcal{C}$ of short Reeb chords. In addition, we define curved augmentations of the LSFT algebra and show that they can be used to construct a homotopy inverse of the $A_\infty$ Sabloff map, together with all higher homotopies. The above results suggest a conjectural recipe for an explicit weak relative Calabi-Yau structure on the quotient $A_\infty$ functor $\pi:\mathcal{A}ug_+\to \mathcal{C}$.