Abstract

I will discuss the mirror symmetry of cluster varieties via the enumeration of non-archimedean analytic curves. We construct a canonical scattering diagram by counting infinitesimal non-archimedean cylinders bypassing the Kontsevich-Soibelman algorithm. We prove adic convergence, ring homomorphism, finite polyhedral approximation, theta function consistency and Kontsevich-Soibelman consistency. Furthermore, we prove a comparison theorem with the combinatorial constructions of Gross-Hacking-Keel-Kontsevich. This has two-fold implications: First it gives a concrete combinatorial way for computing the abstract non-archimedean curve counting in the case of cluster varieties; conversely, we obtain geometric interpretations of various combinatorial constructions and answer several conjectures of GHKK . Joint work with S. Keel.