Potentials for Quivers from Twice-Punctured Surfaces

Definitions and mathematical statement

The supplied question concerns quivers associated with triangulations of closed surfaces of positive genus having exactly two punctures. Under the conventional interpretation, let $\Sigma$ be a compact connected oriented surface without boundary, of genus $g\geq 1$, let $M\subset\Sigma$ consist of exactly two distinct marked points, and let $\tau$ be a triangulation of $(\Sigma,M)$. Write $Q(\tau)$ for its signed adjacency quiver. The record does not explicitly specify orientability, connectedness, the kind of triangulation, or the conventions for self-folded triangles; this interpretation therefore requires source confirmation.

Fix a coefficient field $k$, whose required hypotheses are not supplied. For a finite quiver $Q$ without loops or oriented two-cycles, let $k\langle\!\langle Q\rangle\!\rangle$ denote its completed path algebra, completed with respect to the ideal generated by the arrows. A potential on $Q$ is a possibly infinite $k$-linear combination of cyclic paths, considered up to cyclic equivalence. Here cyclic equivalence identifies a cyclic path with its cyclic rotations and takes the closure of the resulting relations in the arrow-ideal topology.

Two potentials $S,S'$ on $Q$ are weakly right equivalent, written $S\sim_{\mathrm{wr}}S'$, if there are a scalar $\lambda\in k^{\times}$ and a continuous $k$-algebra automorphism $\varphi$ of $k\langle\!\langle Q\rangle\!\rangle$, fixing every vertex idempotent, such that $\varphi(S)$ is cyclically equivalent to $\lambda S'$. A potential is non-degenerate in the quiver-with-potential mutation sense if every finite sequence of vertex mutations can be performed successively and, after reduction at each step, the resulting quiver has no oriented two-cycles. Let $\mathcal P_{\mathrm{nd}}(Q;k)$ denote the collection of such potentials.

Question, with unresolved source conventions. For every surface and triangulation in the intended class, is there exactly one weak right equivalence class of non-degenerate potentials on its associated quiver? With the coefficient field and triangulation conventions fixed, the question is \[ \left|\mathcal P_{\mathrm{nd}}\left(Q(\tau);k\right)/\!\sim_{\mathrm{wr}}\right|=1 \qquad\text{for every admissible }(\Sigma,M,\tau),\quad g(\Sigma)\geq 1,\quad |M|=2. \] This asks both for existence and for equivalence of any two non-degenerate potentials on each fixed quiver. It does not assert equivalence between potentials on different quivers. The coefficient-field hypotheses and the precise class of admissible triangulations are unresolved in the supplied record; the display is a formulation schema, not an assertion universally quantified over all fields.

Short English statement

For every quiver obtained from a triangulation of a closed positive-genus surface with exactly two punctures, ask whether a non-degenerate potential exists and whether any two such potentials differ only by a right equivalence and multiplication by a nonzero scalar. The coefficient field and triangulation conventions require confirmation. The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002450$, title: Potentials for Quivers from Twice-Punctured Surfaces; collection: Oberwolfach Reports Open Problems; exact source problem code: catalogue entry. The record gives proposed year $2013$ and names no proposer.
• Supplied source citation: Cluster Algebras and Related Topics (2014), Oberwolfach Reports, pp. 3379--3432, DOI 10.4171/owr/2013/58. https://doi.org/10.4171/owr/2013/58. This is also the supplied literature-source link labeled "original OWR report." The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature-source attribution: Geiss, Labardini-Fragoso and Schroer, DOI 10.1016/j.aim.2015.09.038. https://doi.org/10.1016/j.aim.2015.09.038. No article title or precise theorem reference is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that uniqueness is proved for many mutation classes, while closed positive-genus surfaces with exactly two punctures remain an unresolved classification case. This is supplied dated curation derived from the OWR material, not an independently verified assessment; the status and cited claims require independent verification.
• The registry labels the extracted statement "exact" and reports that manual review found it self-contained, mathematically coherent, and free of LaTeX or report-layout artifacts. These are supplied verification claims, not verification performed for this draft.