Charting Class ${\cal S}_k$ Territory

TL;DR

This work defines and analyzes class $\mathcal{S}^1_k$, a broad family of 4d $\mathcal{N}=1$ SCFTs within the $\mathcal{S}_k$ program, realized as $\mathbb{Z}_k$ orbifolds of Type IIA brane setups with rotated NS-branes and connected to compactifications of the 6d $(1,0)$ theories $\mathcal{T}_k^N$ on punctured spheres. It develops core theories with two standard orderings, studies Seiberg dualities as puncture exchanges, and uses the superconformal index to verify duality invariances. The paper also introduces a rich set of building blocks (free trinions) and two gluing prescriptions ($\mathcal{N}=2$ and $\mathcal{N}=1$), extends gluing beyond the core theories, and demonstrates closing punctures via Higgsing, all within a bipartite-graph (BFT) framework with zig-zag path diagnostics for global symmetries. Overall, it provides a systematic, geometric and combinatorial toolkit for constructing, classifying, and testing $\mathcal{S}_k$ theories, with implications for dualities, marginal deformations, and connections to D-brane realizations and higher-dimensional SCFTs.

Abstract

We extend the investigation of the recently introduced class ${\cal S}_k$ of 4d $\mathcal{N}=1$ SCFTs, by considering a large family of quiver gauge theories within it, which we denote $\mathcal{S}^1_k$. These theories admit a realization in terms of $\mathbb{Z}_k$ orbifolds of Type IIA configurations of D4-branes stretched among relatively rotated sets of NS-branes. This fact permits a systematic investigation of the full family, which exhibits new features such as non-trivial anomalous dimensions differing from free field values and novel ways of gluing theories. We relate these ingredients to properties of compactification of the 6d (1,0) superconformal ${\cal T}_N^k$ theories on spheres with different kinds of punctures. We describe the structure of dualities in this class of theories upon exchange of punctures, including transformations that correspond to Seiberg dualities, and exploit the computation of the superconformal index to check the invariance of the theories under them.

Figures (24)

• Figure 1: Type IIA brane configuration for core class $\mathcal{S}^1_k$ theories. The configuration involves D4-branes suspended among $n_5$ NS5 and $n_5'$ NS5'-branes, in an arbitrary order, and located at a $\mathbb{C}^2/{\mathbb{Z}}_k$ orbifold. At the bottom we show the associated linear quiver corresponding to the brane configuration before the $\mathbb{Z}_k$ orbifold. Blue circular nodes correspond to gauge symmetries while yellow square nodes correspond to global symmetries. Dashed arrows represent non-dynamical fields in the adjoint representation of the global nodes.
• Figure 2: The two basic types of nodes in the linear quiver. Type I corresponds to D4-branes stretched between an NS5-NS5' pair. Type II corresponds to D4-branes stretched between an NS5-NS5 or an NS5'-NS5' pair.
• Figure 3: Riemann surface for the theory in Figure \ref{['basic_IIA']}. Grey punctures are maximal, and the blue and red punctures correspond to the two kinds of minimal ones.
• Figure 4: The two possible kinds of columns of nodes descending from parent Type I or Type II nodes after orbifolding, in quiver and dimer languages. The top and bottom blue lines are identified, giving the quiver and dimer a cylinder topology.
• Figure 5: The result of orbifolding the global symmetry nodes in the parent theory, in quiver and dimer languages. The non-dynamical dashed arrows endow the maximal puncture with an orientation, in this case ascending.