Positivity for quantum cluster algebras
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.
Cite
@article{arxiv.1601.07918,
title = {Positivity for quantum cluster algebras},
author = {Ben Davison},
journal= {arXiv preprint arXiv:1601.07918},
year = {2017}
}
Comments
v6 - 47 pages, final version, to appear in Annals of Mathematics; v5 - improvements in exposition + minor corrections (thanks to the referee) 46 pages; v4 - Final draft - innumerable corrections and expanded explanations over previous version. 43 pages