A 5d perspective on the compactifications of 6d SCFTs to 4d $\mathcal{N}=1$ SCFTs

TL;DR

This paper explores the compactification of 6d SCFTs to 4d $ abla=1$ theories through a 5d perspective, utilizing duality domain walls and the box graph description to understand properties like flux and global symmetries, exemplified by the rank 1 E-string.

Contribution

It rephrases the 6d to 4d compactification process using 5d duality domain walls and box graphs, providing new insights into flux and symmetry properties from a 5d viewpoint.

Findings

Reformulation of 6d to 4d compactification via 5d domain walls.

Recovery of flux and symmetry properties from the 5d perspective.

Application to the rank 1 E-string theory as an example.

Abstract

Compactifying 6d superconformal field theories (SCFTs) to 4d $\mathcal{N}=1$ theories on two-punctured spheres (tubes) and tori with flux is realized using duality domain walls in 5d $\mathcal{N}=1$ Kaluza-Klein (KK) theories, which are usually denoted by $flux$ $domain$ $walls$. We revisit this construction and study it in detail from the 5d perspective, specifically rephrasing it using the box graph description of the extended Coulomb branch phases of 5d theories. This perspective could be helpful in understanding how to equivalently realize the 4d $\mathcal{N}=1$ models from geometric engineering in M-theory. Along the way, we show how to recover various properties of the 4d theories from the 5d perspective, such as the flux associated to the domain wall configurations and the presence of a $\mathfrak{u}(1)$ global symmetry in the 4d theory descending from the KK symmetry on the tube, which is broken to a discrete subgroup on a flux torus. We demonstrate all of these ideas using the rank 1 E-string theory.