Isomorphism in the augmentation category

TL;DR

This paper establishes a precise relationship between augmentations of Legendrian submanifolds and algebraic transformations, extending known results from knots to higher dimensions using quantum flow trees.

Contribution

It proves that isomorphism of augmentations corresponds to a combination of dga homotopy and dilation, generalizing previous knot/link results to higher dimensions with new techniques.

Findings

Augmentations are isomorphic iff they differ by a dga homotopy and dilation.

The augmentation category is not always equivalent to the microlocal rank 1 sheaf category.

Quantum flow tree techniques are used to extend the theory to higher dimensions.

Abstract

Given a Legendrian submanifold in any dimension, we prove that two augmentations are isomorphic within the positive augmentation category exactly when they differ by a combination of a dga homotopy and a dilation. This extends the corresponding statement for Legendrian knots and links, but instead of relying on the dga for consistent copies, we make use of quantum flow tree techniques. Consequently, we can strengthen and clarify a result of the first author as follows: for knot contact homology, the augmentation category is not in general equivalent to the microlocal rank 1 sheaf category.