Augmentation categories in higher dimensions

TL;DR

This paper extends the construction of the augmentation category, originally developed for 3-dimensional contact manifolds, to higher-dimensional contact manifolds, providing a new algebraic framework for Legendrian submanifolds.

Contribution

It generalizes the augmentation category construction to higher dimensions, broadening its applicability in symplectic and contact topology.

Findings

Constructs the augmentation category over a field of characteristic 2.

Extends the Ng-Rutherford-Shende-Sivek-Zaslow framework to higher dimensions.

Provides a unital $A_$-category for Legendrian submanifolds in higher-dimensional contact manifolds.

Abstract

For an exact symplectic manifold $M$ and a Legendrian submanifold $\Lambda$ 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.