Explicit derived Hall algebras of the 3-Calabi–Yau categories of quivers with potential and their link to the quantized Fock–Goncharov cluster variety

For a finite quiver $Q$ with a generic potential $W$ , the Ginzburg dg algebra $\Gamma(Q,W)$ defines a 3-Calabi–Yau category $\mathcal{T}_Q$ ; Keller–Yang–Zhou (2009) computed the derived Hall algebra of $\mathcal{T}_Q$ only in the rank-one case where $Q$ is a single vertex, identifying its abelianization with the coordinate ring of the one-dimensional cluster torus. The open problem is the general case: give a generators-and-relations presentation of the derived Hall algebra of $\mathcal{T}_Q$ for an arbitrary quiver with potential, construct explicit algebra homomorphisms from it to the quantized coordinate algebra of the Fock–Goncharov cluster variety $\mathcal{X}_Q$ , and identify commutative quotients with coordinate rings of cluster tori, thereby establishing the Hall-algebraic model of quantized cluster varieties envisioned by Kontsevich–Soibelman and Bridgeland. Later work (stability structures and motivic Donaldson–Thomas theory, cohomological Hall algebras, Nagao's cluster transformation formulae for Donaldson–Thomas invariants, Efimov's quantum cluster variables, and 2024–2025 constructions of cohomological Hall algebras for general 3-Calabi–Yau categories) built the surrounding structural theory and special cases but left the explicit presentation and the Hall-algebra-to-cluster-variety identification open.

Background

A quiver $Q$ is a finite directed graph; in cluster theory one assumes no loops and no oriented 2-cycles. Fomin and Zelevinsky's cluster algebra $\mathcal{A}_Q$ is the algebra generated by 'cluster variables' produced by iterated mutations (a prescribed birational exchange operation on the variables) from an initial algebraic torus, and for acyclic $Q$ it has a very good categorical model: the cluster category, a triangulated category built from the derived category of representations of $Q$ , whose cluster-tilting objects — maximal families of objects with vanishing mutual extensions — correspond to the clusters. Fock and Goncharov enlarged this picture to a cluster ensemble: to $Q$ they associated two dual spaces, the $\mathcal{A}$ -variety and the $\mathcal{X}$ -variety $\mathcal{X}_Q$ , obtained by gluing algebraic tori (one per seed, i.e. per cluster reached by mutation) along cluster mutation maps; $\mathcal{X}_Q$ carries a natural Poisson structure (a Lie bracket on its regular functions compatible with multiplication) and a quantization, a deformation of its coordinate ring built from non-commutative quantum tori (algebras generated by invertible elements $x_i$ with relations $x_i x_j = q^{b_{ij}} x_j x_i$ ) linked by quantum dilogarithm factors under mutation (the quantum dilogarithm is the formal power series $\prod_{n\ge 1}(1+v^{2n-1}x)^{-1}$ , where $v$ is the deformation parameter). The Hall algebra of a finitary category (one whose Hom and extension spaces are finite) over a finite field has a basis of isomorphism classes of objects, with structure constants counting short exact sequences; Ringel's theorem connects such algebras of quiver representations to quantum groups (quantized enveloping algebras of semisimple Lie algebras). For triangulated categories the analogous notion is the derived Hall algebra introduced by Toën and, in a variant, by Xiao and Xu: its multiplication counts distinguished triangles and is twisted by powers of the field size coming from the Euler bilinear form. To a quiver $Q$ with a potential $W$ (a linear combination of cyclic paths in $Q$ ) Ginzburg associated a differential graded algebra $\Gamma(Q,W)$ (a graded algebra carrying a degree-one differential compatible with the multiplication) whose category $\mathcal{T}_Q$ of dg modules with finite-dimensional total homology is 3-Calabi–Yau, meaning its suspension functor $\Sigma$ (the shift of the triangulated structure) satisfies $\mathrm{Hom}(X,Y)\cong \mathrm{D}\,\mathrm{Hom}(Y,\Sigma^{3}X)$ , where $\mathrm{D}$ denotes the dual over the ground field; in particular a '3-spherical object' $S$ has graded endomorphisms only $\mathrm{End}^{*}(S,S)\cong k\oplus k[-3]$ . Around 2008, Kontsevich–Soibelman and Bridgeland initiated a program linking Donaldson–Thomas theory (the enumerative theory counting semistable objects in a 3-Calabi–Yau category), Hall algebras of such categories, and cluster transformations. Keller–Yang–Zhou computed the first non-trivial instance: for $Q$ a single vertex with no arrows, $\mathcal{T}_Q$ is the algebraic triangulated category generated by one 3-spherical object; they proved it is unique up to triangle equivalence, classified its objects via a theorem of Jørgensen (applicable because the generator has a hereditary graded endomorphism ring, one whose self-extension groups vanish in degrees two and higher), and gave an explicit generators-and-relations presentation of the derived Hall algebra, extended to a $d$ -spherical object of arbitrary $d$ . In their section 6 they established the link with the cluster variety: the abelianization of the Hall algebra (its quotient by the space of commutators) is $\mathbb{Q}(v)[x^{\pm 1}]$ , the coordinate ring of the one-dimensional torus which is $\mathcal{X}_Q$ in this rank-one case. Their introduction states the gap this problem records: in contrast to the acyclic cluster algebra $\mathcal{A}_Q$ , for the cluster variety $\mathcal{X}_Q$ of a general quiver 'there is, for the moment, no corresponding theory', while ongoing work of Kontsevich–Soibelman and Bridgeland indicated a close link between the quantization of $\mathcal{X}_Q$ and the Hall algebra of $\mathcal{T}_Q$ .

Problem Statement

For a finite quiver $Q$ without loops and 2-cycles that lies in the admissible class defined below, equipped with a generic potential $W$ , let $\mathcal{T}_Q$ be the 3-Calabi–Yau category of dg modules with finite-dimensional total homology over the Ginzburg dg algebra $\Gamma(Q,W)$ , taken over a finite field $\mathbb{F}_q$ with $q$ a prime power as in Keller–Yang–Zhou, and let $H(\mathcal{T}_Q)$ denote its derived Hall algebra over $\mathbb{Q}$ in the Toën/Xiao–Xu sense (basis of isomorphism classes of objects, structure constants the Hall numbers counting distinguished triangles, twisted by powers of $q$ ). Here 'generic' means that $W$ lies in a Zariski-open non-empty locus in a positive-dimensional space of potentials, equivalently a generic gauge-equivalence class, in the sense of Kontsevich–Soibelman (arXiv:0811.2435, sections 8.1 and 8.3, where for a generic potential the quantum Donaldson–Thomas element depends only on the exchange matrix); this makes the admissible class exactly the loop-free, 2-cycle-free quivers with at least one oriented cycle, while an acyclic quiver with arrows has the single-point potential space $\{0\}$ , so its only potential $W=0$ is non-generic and lies outside the admissible class unless the answer extends the theory to them (the arrowless one-vertex case of Keller–Yang–Zhou being the base case to which (i) and (iii) must reduce). The field $\mathbb{F}_q$ (possibly after a finite extension, to be pinned by the answer) is assumed such that $\mathcal{T}_Q$ is an $\mathbb{F}_q$ -linear Hom-finite category with $\mathrm{Hom}(X,\Sigma^{-i}Y)=0$ for all but finitely many $i>0$ , the finiteness hypotheses recalled in Keller–Yang–Zhou's section 5.1 under which the derived Hall algebra with Hall-number structure constants exists. Since Keller–Yang–Zhou's presented algebra is opposite to the Toën/Xiao–Xu convention (their section 5.1), and since $q$ denotes both the field size and, in item (ii), the formal quantum-torus parameter (with $v=\sqrt{q}$ the dilogarithm parameter), an answer must fix one consistent opposite-algebra and parameter convention across all items. Establish the explicit theory linking $H(\mathcal{T}_Q)$ to the quantized cluster variety $\mathcal{X}_Q$ of Fock–Goncharov — generalizing, in (i) and (iii), what Keller–Yang–Zhou did for the one-vertex quiver, and realizing, in (ii), the numerical counterpart of the Kontsevich–Soibelman integration program (arXiv:0811.2435) — namely:

(i) compute and present the derived Hall algebra $H(\mathcal{T}_Q)$ by explicit generators and relations, in the standard finite-presentation sense exemplified by Keller–Yang–Zhou's Theorem 5.1 (finitely many families of generators with effectively verifiable relations), not merely the free algebra modulo the kernel of the canonical map;

(ii) construct algebra homomorphisms from $H(\mathcal{T}_Q)$ tensored with $\mathbb{Q}(v)$ , $v=\sqrt{q}$ — or from the Kontsevich–Soibelman twisted Hall-algebra variant related to the Toën algebra by their Proposition 12 — to the Fock–Goncharov quantization of $\mathcal{X}_Q$ , as the numerical counterpart, for $\mathcal{T}_Q$ over $\mathbb{F}_q$ , of the Kontsevich–Soibelman integration map (arXiv:0811.2435, Theorem 8 and sections 8.3–8.4, where the integration map is a homomorphism of graded algebras from the motivic Hall algebra to the quantum torus, specialized to per-seed quantum tori of a quiver with generic potential and made mutation-compatible via quantum dilogarithm conjugation): the per-seed quantum torus algebras over $\mathbb{Z}[q^{\pm 1}]$ (or with $\mathbb{Q}(v)$ coefficients after specialization of the formal parameter) with relations $q^{-\hat\varepsilon_{ij}}X_iX_j=q^{-\hat\varepsilon_{ji}}X_jX_i$ , glued by the quantum mutation maps $\mu_k^q=\mathrm{Ad}_{\Psi_{q_k}}(X_k)$ given by conjugation by the quantum dilogarithm, the homomorphisms being compatible with all cluster mutations and quantum dilogarithm factors in the sense of Kontsevich–Soibelman's sections 8.3–8.4;

(iii) identify natural commutative quotients of $H(\mathcal{T}_Q)$ (such as the abelianization by commutators, or specializations of the deformation parameter) chartwise with the coordinate rings $\mathbb{Q}(v)[X_i^{\pm 1}]$ of the classical cluster tori glued into $\mathcal{X}_Q$ , compatibly with the cluster mutation maps (the $q=1$ specialization of the quantum mutations), in a way that reduces in the one-vertex case — where there is a single chart — to Keller–Yang–Zhou's identification of the abelianization with $\mathbb{Q}(v)[x^{\pm 1}]$ (their Lemma 6.1);

thereby verifying, or refuting, the conjectural Hall-algebraic model of quantized cluster varieties envisioned in the work of Kontsevich–Soibelman and Bridgeland. These three components jointly serve one objective — the categorical Hall-algebraic model of $\mathcal{X}_Q$ . For a verification outcome, (i), (ii) and (iii) are jointly required. Alternatively, a documented refutation of the conjectural model for one admissible pair $(Q,W)$ is an admissible outcome in place of carrying out (ii)–(iii) for that pair, component (i) remaining required on this branch; such a refutation is a fully detailed, independently verifiable disproof, for that pair, of the existence of the mutation-compatible homomorphism of (ii) or of the chartwise quotient identification of (iii).

References

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