On the compactification of 5d theories to 4d

TL;DR

This paper explores how 5d superconformal field theories can be systematically reduced to 4d theories, identifying parent-child relationships and mapping mass deformation trajectories among rank-2 4d SCFTs.

Contribution

It provides a comprehensive framework for understanding the reduction of 5d SCFTs to 4d, including criteria for moduli space consistency and the mapping of mass deformations.

Findings

Identified nearly all 5d parents of rank-2 4d SCFTs.

Mapped mass deformation trajectories among 4d theories.

Established criteria for moduli space consistency under RG flows.

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.