Fock-Goncharov dual cluster varieties and Gross-Siebert mirrors

TL;DR

This paper connects cluster varieties with Gross-Siebert mirror symmetry, showing that the mirror of an $ ext{X}$ cluster variety is a degeneration of its dual $ ext{A}$ variety, and provides an enumerative interpretation of scattering diagrams.

Contribution

It bridges cluster variety theory with Gross-Siebert mirror symmetry, demonstrating the mirror relationship as degenerations and interpreting scattering diagrams enumeratively.

Findings

Mirror to $ ext{X}$ cluster variety is a degeneration of dual $ ext{A}$ variety.

Established an enumerative interpretation of cluster scattering diagrams.

Proved Frobenius structure conjecture for certain log Calabi-Yau varieties.

Abstract

Cluster varieties come in pairs: for any $\mathcal{X}$ cluster variety there is an associated Fock-Goncharov dual $\mathcal{A}$ cluster variety. 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. In this paper we bridge the theory of cluster varieties with the algebro-geometric framework of Gross-Siebert mirror symmetry. Particularly, we show that the mirror to the $\mathcal{X}$ cluster variety is a degeneration of the Fock-Goncharov dual $\mathcal{A}$ 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. Consequently, we derive an enumerative interpretation of the cluster scattering diagram. Along the way, we prove the Frobenius structure conjecture for a class of log Calabi-Yau varieties obtained as blow-ups of toric varieties.