Theta bases are atomic

TL;DR

This paper proves that theta bases form an atomic basis for cluster algebras under a modified universal positivity condition involving the scattering atlas, confirming a conjecture in a specific setting.

Contribution

It demonstrates that theta bases satisfy the atomic basis conjecture with a new universal positivity definition involving the scattering atlas.

Findings

Theta bases form an atomic basis under the scattering atlas.

The modified universal positivity uniquely characterizes theta functions.

The result extends the understanding of bases in cluster algebras.

Abstract

Fock and Goncharov conjectured that the indecomposable universally positive (i.e., atomic) elements of a cluster algebra should form a basis for the algebra. This was shown to be false by Lee-Li-Zelevinsky. However, we find that the theta bases of Gross-Hacking-Keel-Kontsevich do satisfy this conjecture for a slightly modified definition of universal positivity in which one replaces the positive atlas consisting of the clusters by an enlargement we call the scattering atlas. In particular, this uniquely characterizes the theta functions.