Chiral theories of class S

TL;DR

This work generalizes class ${\cal S}$ constructions to four-dimensional $\mathcal{N}=1$ theories with chiral matter arising from M5-branes on $\mathbb{C}^{2}/\mathbb{Z}_{k}$, compactified on a Riemann surface. It introduces a detailed building-block framework tied to punctures and $U(1)_t$ curvature, and develops gluing (gauging) rules that render dual descriptions manifest and yield IR fixed points across different pants decompositions. The authors classify the blocks, connect them to six-dimensional origins, and analyze Higgsing and anomaly structures to derive central charges, showing consistency across a range of linear and cyclic quivers. The results illuminate how geometric data of the compactification controls anomalies and IR physics, offering a robust approach to constructing and comparing a broad family of chiral $\mathcal{N}=1$ SCFTs. Overall, the paper provides a geometric, anomaly-backed framework to study dualities and IR dynamics in orbifolded class ${\cal S}$ theories.

Abstract

We study a class of four-dimensional N=1 superconformal field theories obtained from the six-dimensional (1,0) theory, on M5-branes on C^2/Z_k orbifold singularity, compactified on a Riemann surface. This produces various quiver gauge theories whose matter contents are chiral. We classify the building blocks associated to pairs-of-pants, and study the gauging of them as the gluing of punctures. The Riemann surface picture makes the duality invariance of the resulting quiver theories manifest: the theories associated to the same Riemann surface flow to the same nontrivial infrared fixed point. We explicitly check this from the 't Hooft anomalies of the global symmetries and central charges.

Figures (20)

• Figure 1: Left: Type IIA brane configuration of ${\cal N}=1$ SQCD with $N_{f}=2N$ flavors. Right: its quiver diagram.
• Figure 2: The orbifold projection of the ${\cal N}=1$ SQCD with $N_{f}=2N$ flavors and the quartic coupling. A circle and a box represent an ${\cal N}=1$$SU(N)$ gauge and an $SU(N)$ flavor symmetries respectively. A line with an arrow represents an ${\cal N}=1$ chiral multiplet in the bifundamental representation of two groups.
• Figure 3: Graphical representation of a basic building block. We will refer to the left and the right building blocks as ${\cal B}^{++}_{+}$ and ${\cal B}^{--}_{-}$ respectively.
• Figure 4: We take a Seiberg dual to the second gauge node in Figure \ref{['fig:SQCD1']}. The arrows corresponding to $Q_{3}$, $\tilde{Q}_{2}$, $q_{2}$ and $\tilde{q}_{2}$ are flipped and the chiral multiplets $\phi_{2}$, $\tilde{\phi}_{1}$, $N$ and $\tilde{N}$ are added.
• Figure 5: The building blocks of the theory in Figure \ref{['fig:SQCD2']}.