Quivers with potentials associated to triangulated surfaces
Daniel Labardini-Fragoso
TL;DR
The paper links cluster algebras from triangulated bordered surfaces to DWZ quivers with potentials by associating to each ideal triangulation a QP (Q(τ), S(τ)) whose mutations mirror flips of the triangulation. It provides a detailed construction of S(τ) from interior triangles and puncture cycles, proves compatibility between triangulation flips and QP-mutations, and analyzes how reduction affects the mutation class. A main result shows that for surfaces with non-empty boundary, these QPs are rigid and thus non-degenerate, yielding finite-dimensional Jacobian algebras and a well-behaved mutation theory across triangulations. The work also clarifies the relationship to existing cluster-tilted and gentle algebras in special cases and outlines conjectures for empty-boundary surfaces. Together, these results deepen the representation-theoretic interpretation of cluster mutations in the geometric setting of 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.