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

TL;DR

The work analyzes 6d D-type orbi-instanton SCFTs and their torus reductions to 4d, showing that a full class S description on a three-punctured sphere with untwisted D-type punctures is not universal across the D-type landscape. It introduces s-labels (Kac-type) and m-labels to encode puncture data and streamline the 6d→4d mapping, and uses 3d magnetics-quiver mirrors to extract excess/balance information that correlates with global symmetries. The paper establishes two principal classes of 6d quivers that admit D-type class S descriptions, analyzes theta-angle distinctions, and provides extensive consistency checks via Coulomb spectra, anomalies, and Schur indices, complemented by explicit D3–D5 examples and a detailed embedding-by-cyclic-groups framework. It also uncovers hidden Higgsings—RG flows present in the 6d parent theories not manifest in the puncture closures of the 4d class S descriptions—and discusses the limits of the class S approach for D-type theories, including several exceptions and special cases with spinor bifundamentals or empty curves. Overall, the work clarifies the relationship between 6d D-type quivers and 4d class S theories, supplies practical labeling schemes to identify corresponding 4d theories, and highlights the nuanced role of theta angles and Higgsing phenomena in the 6d→4d correspondence.

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 $θ$ 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 $\mathcal{S}$ descriptions.

Figures (1)

• Figure 7.1: The moves of the boxes in the puncture. On the left, we have $\mathfrak{M}=2$. On the right, we have $\mathfrak{M}=-2$.