Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 055, 30 pages      arXiv:2103.12232      https://doi.org/10.3842/SIGMA.2022.055

Mirror Symmetry for Truncated Cluster Varieties

Benjamin Gammage a and Ian Le b
a) Department of Mathematics, Harvard University, USA
b) Mathematical Sciences Institute, Australian National University, Australia

Received August 25, 2021, in final form July 15, 2022; Published online July 19, 2022

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.

Key words: homological mirror symmetry; cluster varieties; almost toric fibrations.

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