Quivers with potentials associated to triangulated surfaces, Part II: Arc representations

TL;DR

This paper extends the theory of quivers with potentials from triangulated surfaces by associating explicit representations to arcs and triangulations, linking mutations to geometric flips.

Contribution

It introduces a new explicit construction of representations for quivers with potentials tied to surface triangulations, connecting mutation theory with geometric arc representations.

Findings

Representations are explicitly associated to arcs and triangulations.

Mutations correspond to flips between triangulations.

The construction bridges geometric and algebraic aspects of surface cluster algebras.

Abstract

This paper is a representation-theoretic extension of Part I. It has been inspired by three recent developments: surface cluster algebras studied by Fomin-Shapiro-Thurston, the mutation theory of quivers with potentials initiated by Derksen-Weyman-Zelevinsky, and string modules associated to arcs on unpunctured surfaces by Assem-Brustle-Charbonneau-Plamondon. Modifying the latter construction, to each arc and each ideal triangulation of a bordered marked surface we associate in an explicit way a representation of the quiver with potential constructed in Part I, so that whenever two ideal triangulations are related by a flip, the associated representations are related by the corresponding mutation.