Quivers with potentials associated to triangulated surfaces

TL;DR

This paper establishes a correspondence between triangulations of bordered surfaces with marked points and quivers with potentials, showing that flips in triangulations correspond to mutations in quivers, and proves rigidity for surfaces with boundary.

Contribution

It introduces a method to associate quivers with potentials to surface triangulations, linking cluster algebra structures to quiver mutations.

Findings

Quivers with potentials are associated to each ideal triangulation.

Flips in triangulations correspond to mutations in quivers.

Quivers are shown to be rigid and non-degenerate for surfaces with boundary.

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.