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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.
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We study weak coupling perturbative series in 4d N=2 and 5d N=1 supersymmetric gauge theories with Lagrangians. We prove that the perturbative series of these theories in zero instanton sector are Borel summable for various observables. Our result for 4d $mathcal{N}=2$ case supports an expectation from a recent proposal on a semiclassical realization of infrared renormalons in QCD-like theories, where the semiclassical solution does not exist in N=2 theories and the perturbative series are unambiguous, namely Borel summable. We also prove that the perturbative series in arbitrary number of instanton sector are Borel summable for a wide class of theories. It turns out that exact results can be obtained by summing over the Borel resummations in each number of instanton sector.
Building on recent progress in the study of compactifications of $6d$ $(1,0)$ superconformal field theories (SCFTs) on Riemann surfaces to $4d$ $mathcal{N}=1$ theories, we initiate a systematic study of compactifications of $5d$ $mathcal{N}=1$ SCFTs on Riemann surfaces to $3d$ $mathcal{N}=2$ theories. Specifically, we consider the compactification of the so-called rank 1 Seiberg $E_{N_f+1}$ SCFTs on tori and tubes with flux in their global symmetry, and put the resulting $3d$ theories to various consistency checks. These include matching the (usually enhanced) IR symmetry of the $3d$ theories with the one expected from the compactification, given by the commutant of the flux in the global symmetry of the corresponding $5d$ SCFT, and identifying the spectrum of operators and conformal manifolds predicted by the $5d$ picture. As the models we examine are in three dimensions, we encounter novel elements that are not present in compactifications to four dimensions, notably Chern-Simons terms and monopole superpotentials, that play an important role in our construction. The methods used in this paper can also be used for the compactification of any other $5d$ SCFT that has a deformation leading to a $5d$ gauge theory.
We discuss reductions of general N=1 four dimensional gauge theories on S^2. The effective two dimensional theory one obtains depends on the details of the coupling of the theory to background fields, which can be translated to a choice of R-symmetry. We argue that, for special choices of R-symmetry, the resulting two dimensional theory has a natural interpretation as an N=(0,2) gauge theory. As an application of our general observations, we discuss reductions of N=1 and N=2 dualities and argue that they imply certain two dimensional dualities.
We classify 5d N=1 gauge theories carrying a simple gauge group that can arise by mass-deforming 5d SCFTs and 6d SCFTs (compactified on a circle, possibly with a twist). For theories having a 6d UV completion, we determine the tensor branch data of the 6d SCFT and capture the twist in terms of the tensor branch data. We also determine the dualities between these 5d gauge theories, thus determining the sets of gauge theories having a common UV completion.
We explore the connection of anti-de-Sitter supergravity in six dimensions, based on the exceptional F(4) superalgebra, and its boundary superconformal singleton theory. The interpretation of these results in terms of a D4-D8 system and its near horizon geometry is suggested.
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