Augmentation categories in higher dimensions

For an exact symplectic manifold $M$ and a Legendrian submanifold $Λ$ of the contactification $M\times \mathbb{R}$, we construct the augmentation category (over a field of characteristic 2), a unital $A_\infty$-category whose objects are augmentations of the Chekanov-Eliashberg differential graded algebra. This extends the construction of the augmentation category by Ng-Rutherford-Shende-Sivek-Zaslow to contact manifolds of dimension greater than 3.

💡 Research Summary

The paper develops a unital (A_\infty)-category (\mathsf{Aug}_+(M,\Lambda)) whose objects are augmentations of the Chekanov‑Eliashberg differential graded algebra (DGA) associated to a compact Legendrian submanifold (\Lambda) inside the contactization (V=M\times\mathbb R) of an exact symplectic manifold (M). Working over a field of characteristic two, the construction generalizes the augmentation category previously defined by Ng‑Rutherford‑Shende‑Sivek‑Zaslow for Legendrian links in (\mathbb R^3) to arbitrary dimensions.

The author first reviews the necessary background: Legendrian contact homology, the Chekanov‑Eliashberg DGA, stable tame isomorphism, and the notion of quantum flow trees, which combine holomorphic disks with Morse flow lines on a pair of nearby Legendrian copies. The quantum flow tree formalism provides a multi‑scale description of the moduli spaces needed to define higher (A_\infty) operations.

The core of the paper introduces an auxiliary “big” category (\mathcal B). Objects are pairs ((\epsilon,k)) where (\epsilon) is an augmentation and (k\in\mathbb N). Morphisms exist only when the second index is larger, giving (\mathcal B) a poset‑like structure. The morphism complexes are generated by Reeb chords from (\Lambda) to a positive push‑off together with contributions from quantum flow trees. The differential (m_1) and higher operations (m_k) are defined by counting rigid configurations of (k+1) parallel copies of (\Lambda) equipped with chosen augmentations; the counts are performed using arbitrary perturbations of the copies, without any consistency constraints.

To obtain the desired augmentation category, the author localizes (\mathcal B) at the “identity” morphisms, collapsing all objects ((\epsilon,k)) for a fixed augmentation (\epsilon) into a single object. This localization eliminates the artificial poset direction while preserving the homotopy‑theoretic information. The resulting category (\mathsf{Aug}+(M,\Lambda)) is shown to be independent, up to (A\infty)-equivalence, of the chosen perturbations and of the almost complex structure used to define holomorphic disks. Moreover, it is invariant under Legendrian isotopy, establishing a robust invariant of the Legendrian submanifold.

In the final section the author compares the construction with the “consistent sequence of DGAs” approach of Ng‑Rutherford‑Shende‑Sivek‑Zaslow. When the parallel copies can be perturbed so that they form a consistent sequence, the auxiliary category (\mathcal B) reduces to the one used in the 3‑dimensional theory, and the localized augmentation category coincides with the original (\mathsf{Aug}_+). This provides a new proof of invariance for the earlier construction and demonstrates that the present method subsumes it.

Overall, the paper supplies a flexible, dimension‑independent framework for defining augmentation categories, bypasses the delicate consistency requirements of previous works, and opens the door to applications in higher‑dimensional contact topology, microlocal sheaf theory, and symplectic field theory.