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Fock and Goncharov described a quantization of cluster X -varieties (also known as cluster Poisson varieties) in Fock and Goncharov (Ann. Sci. Éc. Norm. Supér. 42(6), 865–930 2009). Meanwhile, families of deformations of cluster X -varieties were introduced in Bossinger et al. (Compos. Math. 156(10), 2149–2206, 2020). In this paper we show that the two constructions are compatible– we extend the Fock-Goncharov quantization of X -varieties to the families of Bossinger et al. (Compos. Math. 156(10), 2149–2206, 2020). As a corollary, we obtain that these families and each of their fibers have Poisson structures. We relate this construction to the Berenstein-Zelevinsky quantization of A -varieties (Berenstein and Zelevinsky, Adv. Math. 195(2), 405–455, 2005). Finally, inspired by the counter-example to quantum positivity of the quantum greedy basis in Lee, et al. (Proc. Natl. Acad. Sci. 111(27), 9712–9716, 2014), we compute a counter-example to quantum positivity of the quantum theta basis. [ABSTRACT FROM AUTHOR]

We give a construction of generalized cluster varieties and generalized cluster scattering diagrams for reciprocal generalized cluster algebras, the latter of which were defined by Chekhov and Shapiro. These constructions are analogous to the structures given for ordinary cluster algebras in the work of Gross, Hacking, Keel, and Kontsevich. As a consequence of these constructions, we are also able to construct theta functions for generalized cluster algebras, again in the reciprocal case, and demonstrate a number of their structural properties. [ABSTRACT FROM AUTHOR]

We prove that the quantum DT-invariants associated to quivers with genteel potential can be expressed in terms of certain refined counts of tropical disks. This is based on a quantum version of Bridgeland's description of cluster scattering diagrams in terms of stability conditions, plus a new version of the description of scattering diagrams in terms of tropical disk counts. The weights with which the tropical disks are counted are expressed in terms of motivic integrals of certain quiver flag varieties. We also show via explicit counterexample that Hall algebra broken lines do not result in consistent Hall algebra theta functions, i.e., they violate the extension of a lemma of Carl–Pumperla–Siebert from the classical setting. [ABSTRACT FROM AUTHOR]