On the compactification of 5d theories to 4d

We study general properties of the mapping between 5d and 4d superconformal field theories (SCFTs) under both twisted circle compactification and tuning of local relevant deformation and CB moduli. After elucidating in generality when a 5d SCFT reduces to a 4d one, we identify nearly all N = 1 5d SCFT parents of rank-2 4dN = 2 SCFTs. We then use this result to map out the mass deformation trajectories among the rank-2 theories in 4d. This can be done by first understanding the mass deformations of the 5dN = 1 SCFTs and then map them to 4d. The former task can be easily achieved by exploiting the fact that the 5d parent theories can be obtained as the strong coupling limit of Lagrangian theories, and the latter by understanding the behavior under compactification. Finally we identify a set of general criteria that 4d moduli spaces of vacua have to satisfy when the corresponding SCFTs are related by mass deformations and check that all our RG-flows satisfy them. Many of the mass deformations we find are not visible from the corresponding complex integrable systems.