Symplectic Structure on Augmentation Varieties

Abstract: In a recent joint project with H. Gao and L. Shen, we introduce a cluster K2 structure on the augmentation variety of the Chekanov-Eliashberg dga for the rainbow closure of any positive braid with marked point decorations. This cluster K2 structure defines a holomorphic presymplectic structure on the complex augmentation variety. Using a result of Goncharov and Kenyon on surface bipartite graphs, we prove that this holomorphic presymplectic structure becomes symplectic after we reduce the number of marked points to a single marked per link component.

algebraic geometrycombinatoricsdifferential geometrygeometric topologyquantum algebrarepresentation theorysymplectic geometry

Audience: researchers in the topic

Legendrians, Cluster algebras, and Mirror symmetry

Series comments: Schedule
School: January 4–8, 2021
Conference: January 11–15, 2021