MSW-type compactifications of 6d $(1,0)$ SCFTs on 4-manifolds

$\def\d{\mathrm{d}}$ In this work, we study compactifications of $6\d$ $(1, 0)$ SCFTs, in particular those of conformal matter type, on Kähler 4-manifolds. We show how this can be realized via wrapping M5 branes on $4$-cycles of non-compact Calabi–Yau fourfolds with ADE singularity in the fiber. Such compactifications lead to domain walls in $3\d$ $\mathcal{N} = 2$ theories which flow to $2\d N = (0, 2)$ SCFTs. We compute the central charges of such $2\d$ CFTs via $6\d$ anomaly polynomials by employing a particular topological twist along the $4$-manifold. Moreover, we study compactifications on non-compact $4$-manifolds leading to coupled $3\d$-$2\d$ systems. We show how these can be glued together consistently to reproduce the central charge and anomaly polynomial obtained in the compact case. Lastly, we study concrete CFT proposals for some special cases.