Quivers with potentials associated to triangulations of closed surfaces with at most two punctures

TL;DR

This paper classifies non-degenerate potentials for quivers from triangulations of closed surfaces with up to two punctures, revealing infinite possibilities for once-punctured surfaces and uniqueness for twice-punctured ones.

Contribution

It proves the existence of infinitely many non-degenerate potentials for once-punctured surfaces and confirms the uniqueness of such potentials for twice-punctured surfaces, addressing open classification problems.

Findings

Infinite non-degenerate potentials for once-punctured surfaces.

Unique non-degenerate potential for twice-punctured surfaces.

Potentials are stable under flips and QP-mutations.

Abstract

We tackle the classification problem of non-degenerate potentials for quivers arising from triangulations of surfaces in the cases left open by Geiss-Labardini-Schr\"oer. Namely, for once-punctured closed surfaces of positive genus, we show that the quiver of any triangulation admits infinitely many non-degenerate potentials that are pairwise not weakly right-equivalent; we do so by showing that the potentials obtained by adding the 3-cycles coming from triangles and a fixed power of the cycle surrounding the puncture are well behaved under flips and QP-mutations. For twice-punctured closed surfaces of positive genus, we prove that the quiver of any triangulation admits exactly one non-degenerate potential up to weak right-equivalence, thus confirming the veracity of a conjecture of the aforementioned authors.