Orbi-Instantons and Class $\mathcal{S}$ Theories of Type D

TL;DR

This paper explores the reduction of 6d D-type orbi-instanton SCFTs to 4d class S theories, introducing new labels for punctures and revealing complex relationships and hidden flows in the theories.

Contribution

It characterizes D-type punctures using s- and m-labels, linking 6d quivers to 4d class S theories, and uncovers hidden Higgsing phenomena in the reduction process.

Findings

D-type punctures characterized by s- and m-labels

Limited subclass of reductions admit class S description

Discovery of hidden Higgsings in 6d to 4d flows

Abstract

We investigate the landscape of 6d $\mathcal{N}=(1,0)$ D-type orbi-instanton superconformal field theories (SCFTs) and their torus compactifications to four-dimensional class $\mathcal{S}$ 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 $\mathcal{S}$ description on a Riemann sphere with three untwisted D-type punctures constitute only a subset of the full orbi-instanton landscape. For this subclass, 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 $\mathcal{S}$ descriptions. Furthermore, we analyse physical distinctions arising from 6d $\theta$ 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 $\mathcal{S}$ descriptions.