Augmentations are Sheaves
arXiv:1502.04939 · doi:10.2140/gt.2020.24.2149
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.
109 pages; v2: added Legendrian mirror example in section 4.4.4, corrected typos and other minor changes; v3: accepted version
References in corpus (4)
Cited by in corpus (13)
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