Stability conditions on triangulated categories

TL;DR

This paper defines stability conditions on triangulated categories, showing that the space of such conditions forms a topological manifold, providing a new invariant in the mathematical study of these categories.

Contribution

It introduces the concept of stability conditions on triangulated categories and proves that their space is a manifold with a natural topology, linking physics and mathematics.

Findings

The set of stability conditions forms a topological space.

The space of stability conditions is a (possibly infinite-dimensional) manifold.

Provides a new invariant for triangulated categories.

Abstract

This paper introduces the notion of a stability condition on a triangulated category. The motivation comes from the study of Dirichlet branes in string theory, and especially from M.R. Douglas's notion of $\Pi$-stability. From a mathematical point of view, the most interesting feature of the definition is that the set of stability conditions $\Stab(\T)$ on a fixed category $\T$ has a natural topology, thus defining a new invariant of triangulated categories. After setting up the necessary definitions I prove a deformation result which shows that the space $\Stab(\T)$ with its natural topology is a manifold, possibly infinite-dimensional.