A Thurston compactification of the space of stability conditions

TL;DR

This paper introduces a new way to compactify the space of stability conditions on triangulated categories, inspired by Thurston's compactification of Teichmüller space, and explores its geometric properties especially for certain quiver categories.

Contribution

It constructs a Thurston-like compactification of the stability space using maps to infinite projective space and analyzes its properties for specific 2-Calabi--Yau categories.

Findings

The maps are injective with compact closure of their images.

Identifies boundary points analogous to intersection functionals in Teichmüller theory.

Fully analyzes the compactification for A2 and Â1 quivers.

Abstract

We propose compactifications of the moduli space of Bridgeland stability conditions of a triangulated category. Our construction arises from a viewing a stability condition as a metric on the underlying category and is inspired by the Thurston compactification of the Teichm\"uller space of hyperbolic metrics on a surface. The key ingredient in the construction are maps from the stability manifold to an infinite projective space. We prove that, under suitable hypotheses, these maps are injective and their image has a compact closure. We identify a family of points in the boundary that are categorical analogous to the intersection functionals in Teichm\"uller theory. We study in detail the geometry of the resulting compactification for the 2-Calabi--Yau categories of quivers, and fully work out the cases of the \(A_2\) and \(\widehat{A_1}\) quivers. To do so, we carefully examine the dynamics of Harder--Narasimhan multiplicities under auto-equivalences of the category. We introduce a finite automaton to study this dynamics and employ it in our analysis of the \(A_{2}\) and \(\widehat{A_1}\) categories.