Bracelets bases are theta bases

TL;DR

This paper proves the equivalence of bracelet bases and theta bases in skein algebras of marked surfaces, establishing their coincidence in both classical and quantum settings, and explores related positivity and scattering diagram properties.

Contribution

It demonstrates that bracelet bases and theta bases coincide in skein and cluster algebras, including quantum cases, and analyzes their behavior under folding of scattering diagrams.

Findings

Bracelet bases and theta bases are shown to coincide.

Quantum bracelet bases are proven to be theta functions.

Canonical coordinates match quantum theta functions, confirming conjectures.

Abstract

The skein algebra of a marked surface, possibly with punctures, admits the basis of (tagged) bracelet elements constructed by Fock-Goncharov and Musiker-Schiffler-Williams. As a cluster algebra, it also admits the theta basis of Gross-Hacking-Keel-Kontsevich, quantized by Davison-Mandel. We show that these two bases coincide (with a caveat for notched arcs in once-punctured tori). In unpunctured cases, one may consider the quantum skein algebra. We show that the quantized bases also coincide. Even for cases with punctures, we define quantum bracelets for the cluster algebras with coefficients, and we prove that these are again theta functions. On the corresponding cluster Poisson varieties (parameterizing framed $PGL_2$-local systems), we prove in general that the canonical coordinates of Fock-Goncharov, quantized by Bonahon-Wong and Allegretti-Kim, coincide with the associated (quantum) theta functions. Long-standing conjectures on strong positivity and atomicity follow as corollaries. Of potentially independent interest, we examine the behavior of cluster scattering diagrams under folding.