Mutations of Quivers with Potential and Dimer Models

Abstract

In this work, we aim to give a brief exposition to a relationship between mutations of quivers with potential and dimer models. Quiver mutation arose naturally from studying the flips of triangulations of regular \(n\)-gons and are combinatorial in nature. Dimer models can be interpreted as finite bipartite tilings of a compact oriented Riemann surface and arose from statistical physics. The work of Derksen, Weymann, and Zelevinsky provide an approach to lift quiver mutation to an algebraic setting by introducing the notion of a quiver with potential. Every dimer model has a naturally associated quiver with potential known as its dimer quiver. We show that a specific kind of transformation, known as urban renewal, of a dimer model induces mutation of the dimer quiver in a natural way with respect to quiver potential.