Tinkertoys for the Z3-twisted D4 Theory

TL;DR

<3-5 sentence high-level summary> We analyze abelian $\,\mathbb{Z}_3\,$-twists of the $D_4(2,0)$ theory on punctured Riemann surfaces to construct 4D $\mathcal{N}=2$ SCFTs, extending the class-S framework under outer-automorphism twists and highlighting the limitations of non-Abelian twists. The work develops a tinkertoy approach for the twisted theory, catalogs regular punctures, fixtures, and gauge-theory fixtures, and uncovers a new rank-1 SCFT with ${SU(4)}_{14}$ global symmetry. It also identifies duality structures involving the $T_{G_2}$ SCFT and Spin(8) gaugings of the $(E_8)_{12}$ SCFT, and discusses resolving atypical punctures via non-commuting twists, including an explicit 6-punctured-sphere construction and a ramification analysis of the covering map to moduli space. The results advance understanding of twisted class-S theories, reveal new isolated fixed points, and clarify when and how atypical punctures can be resolved within or beyond the commuting-twists framework.

Abstract

Among the simple Lie algebras, $D_4$ is distinguished as the unique one whose group of outer-automorphisms is bigger than $\mathbb{Z}_2$. We study the compactifications of the $D_4$ (2,0) Theory on a punctured Riemann surface, $C$, with outer-automorphism twists around cycles of $C$ lying in $\mathbb{Z}_3\subset \text{Aut}(D_4)= S_3$. The resulting 4D $\mathcal{N}=2$ SCFTs have a number of new and interesting properties. As byproduct, we discover a new rank-1 $\mathcal{N}=2$ SCFT with flavour symmetry group $SU(4)$.