Quadratic differentials as stability conditions: collapsing subsurfaces

TL;DR

This paper establishes a deep connection between the stability conditions of a new class of triangulated categories derived from marked surfaces and moduli spaces of framed quadratic differentials, using exchange graph comparisons.

Contribution

It introduces a novel class of triangulated categories as Verdier quotients of 3-Calabi-Yau categories and links their stability conditions to moduli spaces of quadratic differentials with arbitrary zeros and poles.

Findings

Identification of stability condition spaces with moduli spaces of quadratic differentials

Development of a comparison method for exchange graphs via tilting hearts and flipping angulations

Establishment of a new framework connecting triangulated categories and quadratic differentials

Abstract

We introduce a new class of triangulated categories, which are Verdier quotients 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 differentials 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.