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Augmentations are Sheaves
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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.
Forward citations
Cited by 2 Pith papers
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Legendrian skein algebras and Hall algebras
A natural map from the graded Legendrian skein algebra of a surface to the Hall algebra of its Fukaya category is an isomorphism for disks with marked points and injective for annuli.
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Legendrian DGA Representations and the Colored Kauffman Polynomial
For every Legendrian knot, ungraded n-dimensional representation numbers of its contact homology DGA equal the n-colored Kauffman polynomial specialized at a^{-1}=0.
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