Augmentations are Sheaves
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.
Cite
@article{arxiv.1502.04939,
title = {Augmentations are Sheaves},
author = {Lenhard Ng and Dan Rutherford and Vivek Shende and Steven Sivek and Eric Zaslow},
journal= {arXiv preprint arXiv:1502.04939},
year = {2021}
}
Comments
109 pages; v2: added Legendrian mirror example in section 4.4.4, corrected typos and other minor changes; v3: accepted version