Real variations of stability on K3 categories

TL;DR

This paper explores the rich structure of stability conditions in 2-Calabi-Yau categories linked to preprojective algebras of affine and hyperbolic graphs, introducing real flows to connect Coxeter arrangements with homological algebra.

Contribution

It introduces the concept of real flows to connect Coxeter arrangements with homological algebra, extending the machinery of real variations from affine to hyperbolic settings.

Findings

Many real variations of stability conditions exist for these categories.

The notion of real flows categorifies alcove relations in hyperplane arrangements.

Derived categories with nilpotent cohomology are equivalent to those of nilpotent modules over the preprojective algebra.

Abstract

This paper proves that the 2-Calabi-Yau triangulated category associated with the preprojective algebra of an affine or hyperbolic graph admits many real variations of stability conditions, in the sense of Anno, Bezrukavnikov, and Mirkovi\'{c}. We do this by connecting the Coxeter arrangements of the graph with the homological algebra by introducing the concept of real flows. This categorifies the notion of above and below for alcoves in a hyperplane arrangement and allows us to generalise much of the machinery of real variations for affine arrangements to the hyperbolic setting. In the process, we show that the derived category with nilpotent cohomology is equivalent to the derived category of nilpotent modules over the preprojective algebra.