Bangle functions are the generic basis for cluster algebras from punctured surfaces with boundary

TL;DR

This paper proves that bangle functions form a basis for cluster algebras from punctured surfaces with boundary, linking them to Caldero--Chapoton functions from Jacobian algebras, thus confirming their foundational role.

Contribution

It establishes the equivalence of bangle functions and Caldero--Chapoton functions for punctured surfaces, confirming bangle functions as a basis for the associated cluster algebra.

Findings

Bangle functions coincide with Caldero--Chapoton functions for the given surfaces.

Bangle functions form a basis of the cluster algebra when boundary points are at least two.

The result confirms the conjecture that bangle functions are fundamental in this setting.

Abstract

We prove that for any possibly-punctured surface with non-empty boundary $\mathbf{\Sigma}=(\Sigma, \mathbb{M}, \mathbb{P})$, and any tagged triangulation $T$ of $\mathbf{\Sigma}$ in the sense of Fomin--Shapiro--Thurston, the coefficient-free bangle functions of Musiker--Schiffler--Williams coincide with the coefficient-free generic Caldero--Chapoton functions arising from the Jacobian algebra of the quiver with potential $(Q(T), W(T))$ associated to $T$ by Cerulli Irelli and the second author. When the set of boundary marked points $\mathbb{M}$ has at least two elements, Schr\"oer and the first two authors have shown, relying heavily on results of Mills, Muller and Qin, that the generic coefficient-free Caldero-Chapoton functions form a basis of the coefficient-free (upper) cluster algebra $\mathcal{A}(\mathbf{\Sigma})=\mathcal{U}(\mathbf{\Sigma})$. So, the set of bangle functions proposed by Musiker--Schiffler--Williams over ten years ago is indeed a basis.