In the study of cluster algebras, an important question dating back to the introduction of the theory [S. Fomin andA. Zelevinsky, J. Am. Math. Soc. 15, No. 2, 497–529 (2002;Zbl 1021.16017)] concerns the description of various well-behaved bases. Good properties of a basis for a cluster algebra are that it should contain cluster monomials, and have nonnegative structure constants.
In this paper, the authors construct a set of linearly independent elements of a completion of an upper cluster algebra (of type \(\mathcal{A}\) or \(\mathcal{X}\)). This set consists of theta-functions, and is indexed by a subset \(\Theta\) of the tropical points of the mirror dual cluster variety. The authors christen the subspace spanned by these functions the middle cluster algebra – in many situations, such as in type \(\mathcal{X}\) or in type \(\mathcal{A}\) with principal coefficients, this space is contained in the upper cluster algebra and contains the ordinary one.
A conjecture ofV. V. Fock andA. B. Goncharov [Ann. Sci. Éc. Norm. Supér. (4) 42, No. 6, 865–930 (2009;Zbl 1180.53081)], false in general [M. Gross et al., Algebr. Geom. 2, No. 2, 137–175 (2015;Zbl 1322.14032)], is that the set of all tropical points of the mirror dual cluster variety should index a basis of the upper cluster algebra, and the details of the preceding paragraph are presented here as a corrected version of this conjecture. The authors also give sufficient conditions under which \(\Theta\) does consist of all tropical points, and the middle cluster algebra coincides with the upper one, so that the conjecture is true in its original form. These conditions are technical but implied by cluster-theoretically familiar ones, including acyclicity of the initial quiver, or the existence of a maximal green sequence.
The techniques used are geometric, strongly motivated by log-Calabi-Yau geometry. These include scattering diagrams and broken lines, concepts introduced in earlier work involving the authors [M. Kontsevich andY. Soibelman, Prog. Math. 244, 321–385 (2006;Zbl 1114.14027);M. Gross andB. Siebert, Ann. Math. (2) 174, No. 3, 1301–1428 (2011;Zbl 1266.53074);M. Gross, Adv. Math. 224, No. 1, 169–245 (2010;Zbl 1190.14038)]. Indeed, the structure constants for the theta functions are computed by counting broken lines, the obstruction to the Fock-Goncharov conjecture in general being that these counts can be infinite (either in the calculation of an individual structure constant, or in the sense that infinitely many structure constants involved in computing a product may be non-zero). These geometric techniques turn out to give an extremely powerful perspective on cluster algebras, and the authors show how they can be used to prove deep results such as positivity of the Laurent phenomenon [K. Lee andR. Schiffler, Ann. Math. (2) 182, No. 1, 73–125 (2015;Zbl 1350.13024)]. A precursor to this may be found in work of the first three authors, who gave a proof of the Laurent phenomenon using geometric methods [M. Gross et al., Algebr. Geom. 2, No. 2, 137–175 (2015;Zbl 1322.14032)].

Reviewer: Matthew Pressland (Leeds)

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