Stability conditions and the braid group

TL;DR

This paper constructs stability conditions on certain derived categories related to mirror symmetry, revealing a geometric interpretation of braid group actions and their faithfulness, with implications for Lagrangian stability.

Contribution

It provides the first examples of stability conditions on the A-model side of mirror symmetry where the category isn't derived from an abelian category, and describes the space of stability conditions via braid group actions.

Findings

Connected component of stability conditions is the universal cover of a configuration space in .

Braid group acts as deck transformations, explaining their faithfulness.

Geometric origin for braid group actions and stability of Lagrangians.

Abstract

We find stability conditions ([Do], [Br]) on some derived categories of differential graded modules over a graded algebra studied in [RZ], [KS]. This category arises in both derived Fukaya categories and derived categories of coherent sheaves. This gives the first examples of stability conditions on the A-model side of mirror symmetry, where the triangulated category is not naturally the derived category of an abelian category. The existence of stability conditions, however, gives many such abelian categories, as predicted by mirror symmetry. In our examples in 2 dimensions we completely describe a connected component of the space of stability conditions as the universal cover of the configuration space of $(k+1)$ distinct points with centre of mass zero in $\C$, with deck transformations the braid group action of [KS], [ST]. This gives a geometric origin for these braid group actions and their faithfulness, and axiomatises the proposal for stability of Lagrangians in [Th] and the example proved by mean curvature flow in [TY].