Refined Scattering Diagrams and Theta Functions From Asymptotic Analysis of Maurer-Cartan Equations

We further develop the asymptotic analytic approach to the study of scattering diagrams. We do so by analyzing the asymptotic behavior of Maurer-Cartan elements of a (dg) Lie algebra constructed from a (not necessarily tropical) monoid-graded Lie algebra. In this framework, we give alternative differential geometric proofs of the consistent completion of scattering diagrams, originally proved by Kontsevich-Soibelman, Gross-Siebert, and Bridgeland. We also give a geometric interpretation of theta functions and their wall-crossing. In the tropical setting, we interpret Maurer-Cartan elements, and therefore consistent scattering diagrams, in terms of the refined counting of tropical disks. We also describe theta functions, in both their tropical and Hall algebraic settings, in terms of distinguished flat sections of the Maurer-Cartan-deformed differential. In particular, this allows us to give a combinatorial description of Hall algebra theta functions for acyclic quivers with nondegenerate skew-symmetrized Euler forms.

Automated summary

The study further develops the asymptotic analytic approach to analyzing scattering diagrams, providing alternative differential geometric proofs and a geometric interpretation of theta functions. In the tropical setting, Maurer-Cartan elements are interpreted in terms of the refined counting of tropical disks, and theta functions are described in different settings as distinguished flat sections of the deformed differential. This allows for a combinatorial description of Hall algebra theta functions for specific types of key chains.