Journal of Symplectic Geometry

Volume 20 (2022)

Number 2

Augmentations are sheaves for Legendrian graphs

Pages: 259 – 416



Byung Hee An (Department of Mathematics Education, Teachers College, Kyungpook National University, Daegu, South Korea; and Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang, South Korea)

Youngjin Bae (Department of Mathematics, Incheon National University, Incheon, South Korea)

Tao Su (Department of Mathematics, École Normale Supérieure, Paris, France)


In this article, associated to a (bordered) Legendrian graph, we study and show the equivalence between two categorical Legendrian isotopy invariants: the augmentation category, a unital $A_\infty$-category, which lifts the set of augmentations of the associated Chekanov–Eliashberg DGA, and a DG category of constructible sheaves on the front plane, with micro-support at contact infinity controlled by the (bordered) Legendrian graph. In other words, generalizing [21], we prove “augmentations are sheaves” in the singular case.

Received 16 June 2020

Accepted 26 July 2021

Published 23 December 2022