Quivers with potentials associated to triangulated surfaces

Abstract

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

v3: 44 pages, 57 figures. Prop 29 of v2 generalized to Thm 36, some changes to References. In response to referee's comments: some examples added, more cases verified in proof of Thm 30 (formerly Thm 23). Submitted to Proc. London Math. Soc

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