Quadratic differentials as stability conditions

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.Comment: 123 pages; 38 figures. Version 2: hypotheses in the main results mildly weakened, to reflect improved results of Labardini-Fragoso and coauthors. Version 3: minor changes to incorporate referees' suggestions. This version to appear in Publ. Math. de l'IHE