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. © 2008 London Mathematical Society.