$A_\infty$ Sabloff Duality via the LSFT Algebra

TL;DR

This work develops an $A_\infty$-level enhancement of Sabloff duality for Legendrian contact invariants by leveraging Ng's LSFT algebra in a new composable form. It introduces the positive augmentation category $\mathcal{A}ug_+$ and a short chord quotient $\mathcal{C}$, establishing a strict exact sequence of bimodules and a quasi-isomorphism (the $A_\infty$ Sabloff map) between a negative augmentation bimodule and $\mathcal{A}ug_+$, extended by all higher homotopies via curved augmentations. The paper constructs an explicit $m$-copy LSFT framework to encode disks with multiple positive punctures, proves dualizing lemmas for derivations, and demonstrates that the diagonal bimodule of $\mathcal{A}ug_+$ arises as a mapping cone, linking augmentations to short chord data. A key conceptual takeaway is the emergence of a potential weak relative Calabi–Yau structure on the functor $\pi: \mathcal{A}ug_+ \to \mathcal{C}$, supported by a concrete homotopy-inverse machinery and a pathway to higher coherence in Legendrian invariants. Overall, the results provide a robust algebraic framework connecting LSFT, augmentation categories, and Calabi–Yau structures with explicit higher homotopies, enriching the landscape of Legendrian invariants and their categorical realizations.

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 $π:\mathcal{A}ug_+\to \mathcal{C}$.

Figures (15)

• Figure 1: Reeb signs
• Figure 2: Orientation signs for the Chekanov-Eliashberg differential $\partial$
• Figure 3: Illustrated here are the projections of two generic composable strings with holomorphic corners, which are offset from the link for readability. The labelings at crossings are explained in \ref{['holomorphic corner rmk']}. On the left, we have a composable based string of length $3$, corresponding to the composable word $t_1^{-1}q_1q_2p_3$. On the right, we have a cyclically composable string of length $2$, corresponding to the cyclically composable word $p_2p_1$. Unlike the broken closed strings considered in Ng23, our composable strings are allowed to jump across Reeb chords but not between base points.
• Figure 4: Labeling the quadrants at the crossing $a_j$ by the associated LSFT generators $p_j,q_j$. Note that opposite quadrants share the same label.
• Figure 5: Local picture of the gluing operation for two generic composable strings with holomorphic corners

Theorems & Definitions (156)

• Theorem 1.1: cf. \ref{['Q+- triangle']}
• Theorem 1.2: cf. \ref{['A infty Sabloff thm']}
• Theorem 1.3: cf. \ref{['curvature formula']}
• Theorem 1.4: cf. \ref{['higher homotopy thm']}
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6