On the compactification of 5d theories to 4d

TL;DR

The paper analyzes how $5d$ SCFTs map to $4d$ SCFTs under circle compactification and mass deformations, clarifying when a $5d$ theory reduces to a $4d$ SCFT and identifying the $5d$ parents of nearly all rank-2 $4d$ ${ N}=2$ SCFTs. It develops a framework linking 5d mass deformations and their 4d avatars via moduli-space data, guided by Coulomb-branch stratifications and Higgs-branch flows, and uses brane-web realizations to map RG trajectories across dimensions. A set of general criteria is introduced to constrain 4d moduli spaces for RG flows, and the authors show many mass-deformation paths are not visible from known complex integrable systems, underscoring the richer structure in 4d when lifted from 5d. The work provides a comprehensive map of rank-2 4d ${ N}=2$ SCFTs arising from 5d origins, including twisted compactifications and discrete symmetry twists, and offers concrete consistency checks across multiple families, thereby establishing a robust bridge between 5d and 4d SCFT dynamics with potential applications to predicting new 4d fixed points from higher-dimensional data.

Abstract

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.

Figures (13)

• Figure 1: Graphical depiction of the RG-relations among four dimensional ${\mathcal{N}}=2$ SCFTs. We shade in yellow${\mathcal{N}}=2$ Lagrangian theories, in green${\mathcal{N}}=3$ theories and in blue${\mathcal{N}}=4$ theories. All theories but two ${\mathcal{N}}=3$ theories are labeled by their flavor symmetries.
• Figure 2: Graphical depiction of the behavior of the 5d rank-1 $E_n$ theories under both circle compactification (vertical direction) and mass deformation (horizontal direction) associated to 5d gauge coupling. Superconformal theories are labeled in red.
• Figure 3: The brane web representations for some of the $\mathbb{Z}_3$ symmetric $5d$ SCFTs. (a) shows the case of the $5d$ SCFT whose compactifications gives theory 60, that is the $\mathcal{N}=4$$SU(3)$ theory. (b) shows the case of the $5d$ SCFT whose compactifications gives theory 63, which is one of the $\mathcal{N}=3$ theories. (c) shows the case of the $5d$ SCFT whose compactifications gives theory 46. In all cases we are only interested in the external legs and not in how these are connected to one another, which is instead represented by a black blob. The parenthesis gives the $7$-brane charges of the corresponding $7$-branes, or alternatively, the $5$-brane charges of the $5$-branes ending on them.
• Figure 4: Coulomb branch analysis of the mass deformations among the entries of the $\mathfrak{e}_8-\mathfrak{so}(20)$ series.
• Figure 5: Higgs branch analysis of the mass deformations among the entries of the $\mathfrak{e}_8-\mathfrak{so}(20)$ series.