# Refined scattering diagrams and theta functions from asymptotic analysis of Maurer-Cartan equations

## TL;DR

This paper advances the understanding of scattering diagrams and theta functions through asymptotic analysis of Maurer-Cartan equations, providing geometric interpretations and combinatorial descriptions in tropical and Hall algebra contexts.

## Contribution

It introduces a new asymptotic analytic framework for scattering diagrams and theta functions, with geometric proofs and combinatorial descriptions in tropical and Hall algebra settings.

## Key findings

- Alternative proofs of scattering diagram completion
- Geometric interpretation of theta functions and wall-crossing
- Combinatorial description of Hall algebra theta functions

## Abstract

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 differential graded 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 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 non-degenerate skew-symmetrized Euler forms.