Augmentations are sheaves

Ng, Lenhard¹ ; Rutherford, Dan² ; Shende, Vivek³ ; Sivek, Steven⁴ ; Zaslow, Eric⁵

¹ Department of Mathematics, Duke University, Durham, NC, United States
² Department of Mathematical Sciences, Ball State University, Muncie, IN, United States
³ Department of Mathematics, University of California, Berkeley, Berkeley, CA, United States
⁴ Department of Mathematics, Imperial College London, London, United Kingdom
⁵ Department of Mathematics, Northwestern University, Evanston, IL, United States

Geometry & topology, Tome 24 (2020) no. 5, pp. 2149-2286.

Résumé

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.

Détail

DOI : 10.2140/gt.2020.24.2149

Classification : 53D42, 53D37
Keywords: Legendrian knots, Legendrian contact homology, augmentations, constructible sheaves

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