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 $A_{\infty }$–category. This differs from the nonunital category constructed by Bourgeois and Chantraine (J. Symplectic Geom. 12 (2014) 553–583), 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 Shende, Treumann and Zaslow (Invent. Math. 207 (2017) 1031–1133), 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.