Mirror Symmetry for Truncated Cluster Varieties

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.

v5: Section 5 has been deleted and references to Section 5 have been revised

References in corpus (1)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor

Cited by in corpus (4)

• Fock-Goncharov dual cluster varieties and Gross-Siebert mirrors
• Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces
• Homological Mirror Symmetry for local SYZ singularities
• Local mirror symmetry via SYZ