Quadratic differentials as stability conditions: Collapsing subsurfaces

We introduce a new class of triangulated categories, which are Verdier quo- tients of three-Calabi–Yau categories from (decorated) marked surfaces, and show that its spaces of stability conditions can be identified with moduli spaces of framed quadratic dif- ferentials on Riemann surfaces with arbitrary order zeros and arbitrary higher order poles. A main tool in our proof is a comparison of two exchange graphs, obtained by tilting hearts in the quotient categories and by flipping mixed angulations associated with the quadratic differentials.