Mirror Symmetry for Truncated Cluster Varieties

TL;DR

This paper extends mirror symmetry to truncated cluster varieties in arbitrary dimensions, constructing symplectic mirrors and proving homological mirror symmetry, linking cluster theory and toric geometry.

Contribution

It provides a new mirror construction for truncated cluster varieties in any dimension and establishes homological mirror symmetry for these pairs.

Findings

Constructed symplectic mirrors for truncated cluster varieties.

Proved homological mirror symmetry in arbitrary dimensions.

Connected the construction to toric geometry and cluster theory.

Abstract

In the algebraic setting, cluster varieties were reformulated by Gross-Hacking-Keel as log Calabi-Yau varieties admitting a toric model. Building on work of Shende-Treumann-Williams-Zaslow in dimension 2, we describe the mirror to the GHK construction in arbitrary dimension: given a truncated cluster variety, we construct a symplectic manifold and prove homological mirror symmetry for the resulting pair. We also describe how our construction can be obtained from toric geometry, and we relate our construction to various aspects of cluster theory which are known to symplectic geometers.