Curvature of the space of stability conditions

TL;DR

This paper investigates the geometric curvature properties of the space of Bridgeland stability conditions, demonstrating it is neither CAT(0) nor hyperbolic, and explores the hyperbolicity of pseudo-Anosov functors with applications to entropy bounds.

Contribution

It provides the first analysis of the curvature properties of the stability condition space and classifies pseudo-Anosov functors for curves.

Findings

The space of stability conditions is neither CAT(0) nor hyperbolic.

The quotient space by the complex action is not CAT(0) for the Kronecker quiver.

Pseudo-Anosov functors are hyperbolic, giving lower bounds on entropy.

Abstract

Motivated by the study of the autoequivalence group of triangulated categories via isometric actions on metric spaces, we consider curvature properties (CAT(0), Gromov hyperbolic) of the space of Bridgeland stability conditions with the canonical metric defined by Bridgeland. We then prove that the metric is neither CAT(0) nor hyperbolic, and the quotient metric by the natural $\mathbb{C}$-action is not CAT(0) in case of the Kronecker quiver. Moreover, we also show the hyperbolicity of pseudo-Anosov functors defined by Dimitrov-Haiden-Katzarkov-Kontsevich, which yields the lower-bound of entropy by the translation length. Finally, pseudo-Anosov functors in case of curves have been completely classified.