Fock–Goncharov Dual Cluster Varieties and Gross–Siebert Mirrors

Cluster varieties are algebraic varieties obtained by gluing together complex tori using explicit birational transformations. They play an important role in algebra and geometric representation theory, and have the peculiarity to come in pairs (A,X). On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. I will explain how to bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry and show that the mirror to the X-cluster variety is a degeneration of the Fock–Goncharov dual A-cluster variety and vice versa. To do this, we investigate how the cluster scattering diagram of Gross–Hacking–Keel–Kontsevich compares with the canonical scattering diagram defined by Gross–Siebert to construct mirror duals in arbitrary dimensions. This is joint work with Hulya Arguz.

The synchronous discussion for Pierrick Bousseau’s talk is taking place not in zoom-chat, but at https://tinyurl.com/2022-11-18-pb (and will be deleted after ~3-7 days).