Operation of mutation on Polar quivers

Abstract

In recent times, there has been a lot of interest in the study of quivers, both by mathematicians and theoretical physicists. We introduce a new concept of polar quivers and their mutation. The idea of polar quivers arises from the concept of anomaly free R-charges in theoretical physics. Mutation of polar quivers is build on mutation quivers with potential, which was defined by Derksen, Weyman and Zelevinsky. An R-charge assigns angles to the arrows of a quiver. In a polar quiver we assign angles and positive non-zero integers to vertices and impose conditions equivalent to the anomaly conditions for R-charges. We then establish that mutation of a polar quiver will give a polar quiver if and only if a simple additional condition is satisfied. We use families of quivers linked by mutation, from the work of Stern, as our source of examples. The results of this study have applications in geometry and theoretical physics.