Fock–Goncharov dual cluster varieties and Gross–Siebert mirrors

Abstract

Abstract Cluster varieties come in pairs: for any 𝒳 {\mathcal{X}} cluster variety there is an associated Fock–Goncharov dual 𝒜 {\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 𝒳 {\mathcal{X}} cluster variety is a degeneration of the Fock–Goncharov dual 𝒜 {\mathcal{A}} cluster variety and vice versa. To do this, we investigate how the cluster scattering diagram of Gross, Hacking, Keel and Kontsevich compares with the canonical scattering diagram defined by Gross and 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.