Scattering diagrams, theta functions, and refined tropical curve counts

Abstract

Working over various graded Lie algebras and in arbitrary dimension, we express scattering diagrams and theta functions in terms of counts of tropical curves/disks, weighted by multiplicities given in terms of iterated Lie brackets. Over the tropical vertex group, our tropical curve counts are known to give certain descendant log Gromov-Witten invariants. Working over the quantum torus algebra yields theta functions for quantum cluster varieties, and our tropical description sets up for geometric interpretations of these. As an immediate application, we prove the quantum Frobenius conjecture of Fock and Goncharov. We also prove a refined version of the Carl-Pumperla-Siebert result on consistency of theta functions, and we prove the non-degeneracy of the trace-pairing for the Gross-Hacking-Keel Frobenius structure conjecture.