Augmentations are sheaves

• DOI: 10.2140/gt.2020.24.2149
• Publication Date: 2021-01-03T20:34:25Z
• Authors: Lenhard Ng; Dan Rutherford; Vivek Shende; Steven Sivek; Eric Zaslow
• Keywords: Legendrian knots; math.SG; Geometric Topology (math.GT); 53D42; Geological & Geomatics Engineering; augmentations; 01 natural sciences; Legendrian contact homology; constructible sheaves; 0101 Pure Mathematics; 53D37; Mathematics - Geometric Topology; Mathematics - Symplectic Geometry; FOS: Mathematics; math.GT; Symplectic Geometry (math.SG); 0101 mathematics

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.<br/>109 pages; v2: added Legendrian mirror example in section 4.4.4, corrected typos and other minor changes; v3: accepted version<br/>