Quivers with potentials for surface triangulations

Daniel Labardini-Fragoso

In recent works by Fock-Goncharov, Gekhtman-Shapiro-Vainshtein, Fomin-Shapiro-Thurston and Fomin-Thurston, it has been discovered that the triangulations of a surface with marked points carry a natural cluster algebra structure. In particular, it has been shown that the combinatorial operation of 'flip' on triangulations is compatible with the operation of mutation on the signed adjacency quivers of the triangulations. In this talk we discuss the problem of extending this compatibility to the level of quivers with potentials and their representations, by combinatorially defining a potential on the signed adjacency quiver of each triangulation and "arc representations" of the resulting QP, thus establishing a connection with the mutation theory of quivers with potentials recently initiated by Derksen-Weyman-Zelevinsky.