Quivers with potentials associated to triangulated surfaces

We attempt to relate two recent developments: cluster algebras associated to triangulations of surfaces by Fomin–Shapiro–Thurston, and quivers with potentials (QPs) and their mutations introduced by Derksen–Weyman–Zelevinsky. To each ideal triangulation of a bordered surface with marked points, we associate a QP, in such a way that whenever two ideal triangulations are related by a flip of an arc, the respective QPs are related by a mutation with respect to the flipped arc. We prove that if the surface has non‐empty boundary, then the QPs associated to its triangulations are rigid and hence non‐degenerate.

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