On triangulations, quivers with potentials and mutations

TL;DR

This survey explores the relationship between surface triangulations, quivers with potentials, their mutations, and applications in representation theory and physics, highlighting recent advances and open questions.

Contribution

It provides a comprehensive overview of how quivers with potentials derived from surface triangulations relate to mutations, representation types, and triangulated categories in mathematics and physics.

Findings

Mutations of quivers with potentials align with triangulation flips.

Recent results on the representation type of Jacobian algebras.

Uniqueness of non-degenerate potentials established.

Abstract

In this survey article we give a brief account of constructions and results concerning the quivers with potentials associated to triangulations of surfaces with marked points. Besides the fact that the mutations of these quivers with potentials are compatible with the flips of triangulations, we mention some recent results on the representation type of Jacobian algebras and the uniqueness of non-degenerate potentials. We also mention how the the quivers with potentials associated to triangulations give rise to CY2 and CY3 triangulated categories that have turned out to be useful in the subject of stability conditions and in theoretical physics.