Positivity for quantum cluster algebras

TL;DR

This paper proves the positivity of quantum cluster coefficients in skew-symmetric quantum cluster algebras, confirming several longstanding conjectures and connecting to deep structures in algebraic geometry and category theory.

Contribution

It establishes the positivity of quantum cluster coefficients for all skew-symmetric quantum cluster algebras, confirming conjectures by Kontsevich and Efimov.

Findings

Proves positivity of quantum cluster coefficients.

Confirms the Lefschetz property conjecture.

Supports the classical positivity conjecture of Fomin and Zelevinsky.

Abstract

Building on work by Kontsevich, Soibelman, Nagao and Efimov, we prove the positivity of quantum cluster coefficients for all skew-symmetric quantum cluster algebras, via a proof of a conjecture first suggested by Kontsevich on the purity of mixed Hodge structures arising in the theory of cluster mutation of spherical collections in 3-Calabi-Yau categories. The result implies positivity, as well as the stronger Lefschetz property conjectured by Efimov, and also the classical positivity conjecture of Fomin and Zelevinsky, recently proved by Lee and Schiffler. Closely related to these results is a categorified "no exotics" type theorem for cohomological Donaldson-Thomas invariants, which we discuss and prove in the appendix.