Phantom categories on 3-vertex directed DG quivers

Abstract

We construct infinitely many pairwise non-derived-Morita-equivalent phantom categories ${\mathcal{P}}_m,m\in \mathbb{Z}$ from concrete directed DG quivers with 3 vertices. Moreover, there exists a fixed finite-dimensional DG algebra $C$ such that for each $m\in \mathbb{Z}$, there is a way to reassign cohomological gradings on $C$ (without changing the differential) to obtain a new DG algebra $C_m$ with $\mathrm{Perf}(C_m)\cong {\mathcal{P}}_m$.