On the compactification of 5d theories to 4d

We study general properties of the mapping between 5$d$ and 4$d$ superconformal field theories (SCFTs) under both twisted circle compactification and tuning of local relevant deformation and CB moduli. After elucidating in generality when a 5$d$ SCFT reduces to a 4$d$ one, we identify nearly all $\mathcal{N}=1$ 5$d$ SCFT parents of rank-2 4$d$ $\mathcal{N}=2$ SCFTs. We then use this result to map out the mass deformation trajectories among the rank-2 theories in 4$d$. This can be done by first understanding the mass deformations of the 5$d$ $\mathcal{N}=1$ SCFTs and then map them to 4$d$. The former task can be easily achieved by exploiting the fact that the 5$d$ 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 4$d$ 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.