$A_{\infty }$ Sabloff Duality via the LSFT Algebra

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 $\mathcal{A}u{g}_{+}$. We also extend the Ekholm-Etnyre-Sabloff exact sequence to an exact sequence of $\mathcal{A}u{g}_{+}$-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 $\ddot{I}€:\mathcal{A}u{g}_{+}\to \mathcal{C}$.

57 pages, 15 figures