Orbi-instantons and class 𝒮 theories of type D

A bstract We investigate the landscape of 6d 𝒩 = (1 , 0) D-type orbi-instanton superconformal field theories (SCFTs) and their torus compactifications to four-dimensional class 𝒮 theories. By analysing a general class of 6d F-theory constructions via generalised quivers, we demonstrate that — in contrast to the well-characterised A-type series — the dimensional reductions that admit a 4d class 𝒮 description on a Riemann sphere with three untwisted D-type punctures constitute only a subset of the full orbi-instanton landscape. For this sub-class, we show that the punctures can be effectively characterised by two sets of integers: the s -labels and the m -labels. The s -labels, or “Kac-type labels”, serve as the D-type analogues to the Kac labels used in A-type theories; we establish their correspondence with “modified excess numbers” in the associated 3d mirror theories (magnetic quivers). The m -labels are further introduced to streamline the mapping from 6d generalised quivers to their class 𝒮 descriptions. Furthermore, we analyse physical distinctions arising from 6d θ angles and explore the hierarchy of Higgs branch flows. In doing so, we uncover instances of “hidden Higgsings” — renormalization group flows present in the 6d parent theories that are not manifest in the puncture closures of the corresponding class 𝒮 descriptions.