Perverse schobers, stability conditions and quadratic differentials

TL;DR

This paper introduces a unified framework connecting stability conditions of triangulated categories from weighted marked surfaces with moduli spaces of quadratic differentials, using perverse schobers and their global sections.

Contribution

It develops a novel approach based on perverse schobers to identify stability spaces with moduli spaces, generalizing previous results to arbitrary singularity types.

Findings

Identifies spaces of stability conditions with moduli spaces of quadratic differentials.

Establishes a correspondence between mixed-angulations and hearts in triangulated categories.

Generalizes Bridgeland--Smith results to broader classes of quadratic differentials.

Abstract

We develop a unified approach for identifying spaces of stability conditions of triangulated categories arising from weighted marked surfaces with moduli spaces of quadratic differentials. This approach is based on the notion of a perverse schober (perverse sheaf of triangulated categories) and their triangulated categories of global sections. Under suitable conditions on the perverse schober, we identify mixed-angulations and their flips with finite-length hearts and their tilts, which then leads to the identification of moduli spaces. As an application we obtain a generalization of the results of Bridgeland--Smith to quadratic differentials with arbitrary singularity type (zero/pole/exponential).