Fusion-stable structures on triangulated categories

TL;DR

This paper explores fusion-stable structures on triangulated categories, demonstrating their deformation properties and applications to cluster exchange graphs and hyperplane arrangements, offering new insights into Coxeter-Dynkin types and stability conditions.

Contribution

It introduces the concept of fusion-stable structures on triangulated categories and proves a deformation theorem for these stability conditions, connecting them to Coxeter-Dynkin types and hyperplane arrangements.

Findings

Fusion-stable cluster exchange graphs can be realized as categorifications.

Universal covers of Coxeter-Dynkin hyperplane arrangements relate to fusion-stable stability conditions.

Provides an alternative proof of the $K( au,1)$-conjecture for finite Coxeter-Dynkin types.

Abstract

Let $\mathcal{G}$ be a fusion category acting on a triangulated category $\mathcal{D}$, in the sense that $\mathcal{D}$ is a $\mathcal{G}$-module category. Our motivation example is fusion-weighted species, which is essentially Heng's construction. We study $\mathcal{G}$-stable tilting, cluster and stability structures on $\mathcal{D}$. In particular, we prove the deformation theorem for $\mathcal{G}$-stable stability conditions. A first application is that Duffield-Tumarkin's categorification of cluster exchange graphs of finite Coxeter-Dynkin type can be naturally realized as fusion-stable cluster exchange graphs. Another application is that the universal cover of the hyperplane arrangements of any finite Coxeter-Dynkin type can be realized as the space of fusion-stable stability conditions for certain ADE Dynkin quiver. This provides an alternative uniform proof of $K(\pi,1)$-conjecture in the finite Coxeter-Dynkin case.