More rings to rule them all: Fragmentation, 4D ↔ 5D and split-spectral flows

In this note we set-up an explicit 5D construction of AdS-fragmentation, whereby a single black ring splits-up into a multi-black ring configuration. Furthermore it is seen that these fragmented rings are equivalent to a direct 5D lift of 4D multi-center black holes. Along the way we also determine the 4D/5D transformations relevant for multi-center charges. It is seen that the physical charges involved in black ring fragmentation are Page charges arising due to 5D Chern-Simons terms. As an application of these methods, we reproduce the total angular momentum of concentric black rings, originally due to Gauntlett and Gutowski. Finally we provide a geometric interpretation of fragmented black rings using the idea of split-spectral flows, which seeks to study charge shifts of a given black ring due to fluxes generated in a multi-ring background.