Quadratic differentials as stability conditions

TL;DR

This paper establishes a correspondence between moduli spaces of meromorphic quadratic differentials on Riemann surfaces and stability conditions on certain CY3 categories, linking geometric trajectories to stable objects.

Contribution

It introduces a novel connection between quadratic differentials and stability conditions on CY3 categories via quivers with potential.

Findings

Moduli spaces of quadratic differentials are identified with stability condition spaces.

Finite-length trajectories correspond to stable objects in the categories.

Provides a geometric interpretation of stability conditions in terms of quadratic differentials.

Abstract

We prove that moduli spaces of meromorphic quadratic differentials with simple zeroes on compact Riemann surfaces can be identified with spaces of stability conditions on a class of CY3 triangulated categories defined using quivers with potential associated to triangulated surfaces. We relate the finite-length trajectories of such quadratic differentials to the stable objects of the corresponding stability condition.