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We consider the torus compactifications with flux of a class of $6d$ $(1,0)$SCFTs that can be engineered as the low-energy theories on M$5$-branes near anM$9$-plane on a $C^2/Z_2$ singularity. Specifically, we concentrate on the twoSCFTs where the $Z_2$ orbifold action acts non-trivially on the $E_8$ globalsymmetry. We analyze this problem by compactifying to $5d$, where we canexploit the relation to $5d$ duality domain walls. By a suitable guess of thedomain wall theories, the resulting $4d$ theories can be conjectured. These canthen be tested by comparing their properties, notably anomalies and symmetries,against the $6d$ expectations. These constructions lead to various interesting$4d$ phenomena like dualities and symmetry enhancements.

On the torus compactifications of $Z_2$ orbifolds of E-string theories