Abstract
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 mathcal{N} = 1 5d SCFT parents of rank-2 4d mathcal{N} = 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 5d mathcal{N} = 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.
Highlights
Introduction and summary of the resultsIn recent years there has been an incredible progress in understanding superconformal field theories (SCFTs) in various dimensions, those with eight or more supercharges
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
Using a variety of techniques, e.g. matching moduli space of vacua and global symmetries or studying the behavior under gauging of various flavor symmetry factors, we identify the N = 1 5d parent of most known rank-2 4d N = 2 SCFTs, the results are reported in table 1, 2 and 3
Summary
In recent years there has been an incredible progress in understanding superconformal field theories (SCFTs) in various dimensions, those with eight or more supercharges. To obtain an interacting fixed point in 4d, we usually have to tune other dimensionful quantities, e.g. some moduli or relevant deformations, with result that the four dimensional theory obtained heavily depends on the details of how these limits are taken. Despite these challenges, in the absence of twist, we will be able to depict a somewhat coherent picture where specific types of 5d SCFT always reduce to 4d SCFTs. for twisted compactifications the analysis is far less straightforward and we will fall short in providing general conjectures. Lower case bold letter, will instead indicate the global flavor symmetry group (again we will be somewhat sloppy on questions regarding the global structure), e.g. su(3) will denote a theory in 5 or 4 d with global flavor symmetry su(3)
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