Cluster exchange groupoids and framed quadratic differentials

TL;DR

This paper introduces a new mathematical structure called the cluster exchange groupoid for quivers with potential, linking it to triangulations, quadratic differentials, and stability conditions in a way that reveals the simply connected nature of the associated stability space.

Contribution

It constructs the cluster exchange groupoid for quivers with potential and relates its universal cover to a space of framed quadratic differentials, connecting it to stability conditions.

Findings

The universal cover of the groupoid can be constructed via decorated triangulation graphs.

The space of stability conditions is shown to be simply connected.

Relations in the covering groupoid are homotopically trivial in the stability space.

Abstract

We introduce the cluster exchange groupoid associated to a non-degenerate quiver with potential, as an enhancement of the cluster exchange graph. In the case that arises from an (unpunctured) marked surface, where the exchange graph is modelled on the graph of triangulations of the marked surface, we show that the universal cover of this groupoid can be constructed using the covering graph of triangulations of the surface with extra decorations. This covering graph is a skeleton for a space of suitably framed quadratic differentials on the surface, which in turn models the space of Bridgeland stability conditions for the 3-Calabi-Yau category associated to the marked surface. By showing that the relations in the covering groupoid are homotopically trivial when interpreted as loops in the space of stability conditions, we show that this space is simply connected.