Sheaves via augmentations of Legendrian surfaces

Abstract

Given an augmentation for a Legendrian surface in a $1$-jet space, $\Lambda \subset J^1(M)$, we explicitly construct an object, $\mathcal{F}\in S{h}_{\Lambda }$, of the (derived) category from arXiv:1402.0490 of constructible sheaves on $M\times R$ with singular support determined by $\Lambda$. In the construction, we introduce a simplicial Legendrian DGA (differential graded algebra) for Legendrian submanifolds in $1$-jet spaces that, based on arXiv:1608.02984 and arXiv:1608.03011, is equivalent to the Legendrian contact homology DGA in the case of Legendrian surfaces. In addition, we extend the approach of arXiv:1402.0490 for $1$-dimensional Legendrian knots to obtain a combinatorial model for sheaves in $S{h}_{\Lambda }$ in the $2$-dimensional case.

arXiv comment37 pages, 8 figures

References in corpus (11)

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• Microlocal Morse theory of wrapped Fukaya categories
• A complete knot invariant from contact homology
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Cited by in corpus (5)

• Microlocal Morse theory of wrapped Fukaya categories
• Augmentations are sheaves for Legendrian graphs
• Sheaf quantization and intersection of rational Lagrangian immersions
• Augmentations and sheaves for links
• Lagrangian cobordism functor in microlocal sheaf theory I