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Positivity for quantum cluster algebras. (English) Zbl 1408.13055

In the paper under review, the author proves the famous positivity conjecture for quantum cluster algebras associated to quivers. In turn, this also proves the positivity conjecture for skew-symmetric commutative cluster algebras (previously proved by K. Lee and R. Schiffler [Ann. Math. (2) 182, No. 1, 73–125 (2015; Zbl 1350.13024)]).
The proof of the conjecture presented by the author is difficult and beautiful at the same time as it involves a large variety of techniques including Donaldson-Thomas theory, mixed Hodge modules, vanishing cycle cohomology, additive categorification of cluster algebras and Bridgeland stability conditions, among others. In particular, the whole argument relies on a remarkable relation that exists between Donaldson-Thomas theory and (quantum) cluster mutation previously developed by M. Kontsevich and Y. Soibelman [Commun. Number Theory Phys. 5, No. 2, 231–352 (2011; Zbl 1248.14060)], K. Nagao [Duke Math. J. 162, No. 7, 1313–1367 (2013; Zbl 1375.14150)], B. Davison and S. Meinhardt [Geom. Topol. 19, No. 5, 2535–2555 (2015; Zbl 1430.14105)].
We highlight two important points of the proof: firstly, recall that a potential on a quiver \(Q\) is an infinite sum of cyclic words on \(Q\). A potential is algebraic if it is in fact a finite sum. After mutation, an algebraic potential might be transformed into a non-algebraic potential. This is a technical difficulty from the perspective of Donaldson-Thomas theory. In this paper the author presents a remedy for this situation. Secondly, the coefficients of the generators of a quantum cluster algebra sit inside a quantum space \(\mathrm{A}_Q\) (see Section 2.1 of the article under review). The author argues that theory of monodromic Hodge structures developed by Kontsevich-Soibelman allows to categorify \(\hat{\mathrm{A}}_Q\) (a completed version of the quantum space \(\mathrm{A}_Q\)). To prove this the author introduces an integration map from the Grothendieck group of monodromic mixed Hodge structures over the space of dimension vectors to \(\hat{\mathrm{A}}_Q\). This allows to deduce the quantum positivity conjecture from statements about monodromic Hodge modules (for details see Section 3.3 of the article under review).

MSC:

13F60 Cluster algebras
16G20 Representations of quivers and partially ordered sets
16T30 Connections of Hopf algebras with combinatorics
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