Augmentations are Sheaves

TL;DR

This paper demonstrates that the set of augmentations of the Chekanov-Eliashberg algebra forms a sheaf of unital A-infinity categories over the front plane, confirming a conjecture that relates augmentations to sheaves with specific singular support.

Contribution

It establishes that the augmentation category is a sheaf over the x-line and proves its equivalence to a category of sheaves, confirming a conjecture about geometric origins of augmentations.

Findings

Augmentation category forms a sheaf over the x-line.

Confirmed the conjecture relating augmentations to sheaves with singular support.

Every augmentation arises from geometric data.

Abstract

We show that the set of augmentations of the Chekanov-Eliashberg algebra of a Legendrian link underlies the structure of a unital A-infinity category. This differs from the non-unital category constructed in [BC], but is related to it in the same way that cohomology is related to compactly supported cohomology. The existence of such a category was predicted by [STZ], who moreover conjectured its equivalence to a category of sheaves on the front plane with singular support meeting infinity in the knot. After showing that the augmentation category forms a sheaf over the x-line, we are able to prove this conjecture by calculating both categories on thin slices of the front plane. In particular, we conclude that every augmentation comes from geometry.