Department of Mathematics

Seminar in Cluster algebras

  •  Daping Weng, MSU
  •  Augmentations, Fillings, and Clusters of Positive Braid Closures,
  •  07/02/2020
  •  9:30 AM - 10:30 AM

Consider the standard contact structure on $\mathbb{R}^3_{xyz}$ with contact 1-form $\alpha=dz-ydx$. A Legendrian link $\Lambda$ is a link in $\mathbb{R}^3$ along which $\alpha$ vanishes. Chekanov associated a dga (also known as the Chekanov-Eliashberg dga) to every Legendrian link $\Lambda$ and proved that the stable-tame equivalence class of such dga is invariant under Legendrian isotopy of Legendrian links. The augmentation variety of $\Lambda$ is defined to be the moduli space of augmentations of its CE dga. Let $\mathbb{R}^4_{xyzt}$ be the symplectization of the standard contact $\mathbb{R}^3_{xyz}$ and consider the symplectic field theory of exact Lagrangian cobordisms between Legendrian links placed at different constant $t$ slices. Ekholm, Honda, Kalman constructed a contravariant functor that maps an exact Lagrangian cobordism to a dga homomorphism between the corresponding CE dga’s associated to the Legendrian links at the two ends. This induces a covariant functor that maps exact Lagrangian cobordisms to morphisms between augmentation varieties. In the special case of an exact Lagrangian filling of a Legendrian link $\Lambda$, i.e., an exact Lagrangian cobordism from the empty link to $\Lambda$, such covariant functor defines an algebraic torus inside the augmentation variety of $\Lambda$. In this talk, I will focus on the cases of rainbow closures for positive braids, which are naturally Legendrian links. By using an enhanced version of the CE dga, my collaborators and I construct a cluster $K_2$ structure on the augmentation variety of the rainbow closure $\Lambda_\beta$ for any positive braid $\beta$, and prove that the algebraic tori arisen from exact Lagrangian fillings constructed by pinching crossings are cluster charts in this cluster structure. We also relate the Kalman automorphism to the cluster Donaldson-Thomas transformation on these augmentation varieties. As an application of this new cluster structure, we give a sufficient condition on which $\Lambda_\beta$ admits infinitely many non-Hamiltonian-isotopic exact Lagrangian fillings based on the order of DT, solving a conjecture on the existence of Legendrian links that admit infinitely many fillings. This is joint work in progress with Honghao Gao and Linhui Shen.



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