Beschreibung:
<jats:title>A<jats:sc>bstract</jats:sc>
</jats:title><jats:p>We study general properties of the mapping between 5<jats:italic>d</jats:italic> and 4<jats:italic>d</jats:italic> 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<jats:italic>d</jats:italic> SCFT reduces to a 4<jats:italic>d</jats:italic> one, we identify nearly all <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathcal{N} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>N</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> = 1 5<jats:italic>d</jats:italic> SCFT parents of rank-2 4<jats:italic>d</jats:italic><jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathcal{N} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>N</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> = 2 SCFTs. We then use this result to map out the mass deformation trajectories among the rank-2 theories in 4<jats:italic>d</jats:italic>. This can be done by first understanding the mass deformations of the 5<jats:italic>d</jats:italic><jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathcal{N} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>N</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> = 1 SCFTs and then map them to 4<jats:italic>d</jats:italic>. The former task can be easily achieved by exploiting the fact that the 5<jats:italic>d</jats:italic> 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<jats:italic>d</jats:italic> 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.</jats:p>