Ultra diffuse galaxies (UDGs) reveal extreme properties. Here we compile the largest study to date of 85 globular cluster (GC) systems around UDGs in the Coma cluster, using new deep ground-based imaging of the known UDGs and existing imaging from the Hubble Space Telescope of their GC systems. We find that the richness of GC systems in UDGs generally exceeds that found in normal dwarf galaxies of the same stellar mass. These GC-rich UDGs imply halos more massive than expected from the standard stellar mass-halo mass relation. The presence of such overly massive halos presents a significant challenge to the latest simulations of UDGs in cluster environments. In some exceptional cases, the mass in the GC system is a significant fraction of the stellar content of the host galaxy. We find that rich GC systems tend to be hosted in UDGs of lower luminosity, smaller size and fainter surface brightness. Similar trends are seen for normal dwarf galaxies in the Coma cluster. A toy model is presented in which the GC-rich UDGs are assumed to be `failed galaxies within massive halos that have largely old, metal-poor, alpha-element enhanced stellar populations. On the other hand, GC-poor UDGs are more akin to normal, low surface brightness dwarfs that occupy less massive dark matter halos. Additional data on the stellar populations of UDGs with GC systems will help to further refine and test this simplistic model.
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