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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., Comment: 50 pages, 16 figures

We consider compactifications of rank $Q$ E-string theory on a genus zero surface with no punctures but with flux for various subgroups of the $\text{E}_8\times \text{SU}(2)$ global symmetry group of the six dimensional theory. We first construct a simple Wess-Zumino model in four dimensions corresponding to the compactification on a sphere with one puncture and a particular value of flux, the cap model. Using this theory and theories corresponding to two punctured spheres with flux, one can obtain a large number of models corresponding to spheres with a variety of fluxes. These models exhibit interesting IR enhancements of global symmetry as well as duality properties. As an example we will show that constructing sphere models associated to specific fluxes related by an action of the Weyl group of $\text{E}_8$ leads to the S-confinement duality of the $\text{USp}(2Q)$ gauge theory with six fundamentals and a traceless antisymmetric field. Finally, we show that the theories we discuss possess an $\text{SU}(2)_{\text{ISO}}$ symmetry in four dimensions that can be naturally identified with the isometry of the two-sphere. We give evidence in favor of this identification by computing the `t Hooft anomalies of the $\text{SU}(2)_{\text{ISO}}$ in 4d and comparing them with the predicted anomalies from 6d., 57 pages, 6 figures; v2: a few minor changes, published in SciPost Phys