Positivity for quantum cluster algebras from orbifolds

TL;DR

This paper proves positivity for quantum cluster algebras derived from orbifolds by providing combinatorial formulas for quantum Laurent expansions, applicable to any seed, coefficients, and quantization.

Contribution

It introduces combinatorial formulas for quantum Laurent expansions in orbifold-based quantum cluster algebras, establishing positivity.

Findings

Positivity of quantum cluster algebra $ ext{A}_v$ is proved.

Provides explicit combinatorial formulas for quantum Laurent expansions.

Applicable to arbitrary seeds, coefficients, and quantizations.

Abstract

Let $(S,M,U)$ be a marked orbifold with or without punctures and let $\mathcal A_v$ be a quantum cluster algebra from $(S,M,U)$ with arbitrary coefficients and quantization. We provide combinatorial formulas for quantum Laurent expansion of quantum cluster variables of $\mathcal A_v$ concerning an arbitrary quantum seed. Consequently, the positivity for the quantum cluster algebra $\mathcal A_v$ is proved.