Symplectic Structures 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 naturally equips the complex augmentation variety with a holomorphic presymplectic 2-form. Using a result of Goncharov and Kenyon on surface bipartite graphs, we prove that this holomorphic presymplectic 2-form becomes symplectic after we reduce the number of marked points to a single marked per link component (plus some modification).

mathematical physicscommutative algebraalgebraic geometrycombinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

Online Cluster Algebra Seminar (OCAS)