Emergent $\mathcal N=3$ and Frozen $\mathcal N=2$ SCFTs from $\mathcal N=1$ Gauge Theory

Abstract: Four-dimensional $\mathcal{N}=3$ superconformal field theories (SCFTs) are an exotic class of quantum field theories that admit no conventional Lagrangian description. In this Letter, we construct an $\mathcal{N}=1$ supersymmetric gauge theory and argue that its infrared fixed point exhibits supersymmetry enhancement to an $\mathcal{N}=3$ SCFT. The ultraviolet theory has gauge group $SU(2)<sup>3$, a set of chiral multiplets, and a suitable superpotential. By performing $a$-maximization, we obtain infrared central charges $a=c=5/4$, matching those of the rank-one $\mathcal{N}=3$ $\mathbb{Z}_3$ S-fold SCFT. The ultraviolet gauge-theory description allows us to compute the full superconformal index with four independent fugacities. We then examine various limits of the full index and further propose a closed-form expression for the Macdonald index, reproducing all known results for the proposed $\mathcal{N}=3$ theory. More remarkably, we argue that further gauging the $\mathcal{N}=3$ theory yields the frozen rank-one $\mathcal{N}=2$ SCFT conjectured from Seiberg--Witten geometry, providing strong evidence for its existence and unlocking its previously inaccessible full superconformal index. Our construction thus opens a systematic route to exploring the refined protected structure of these exotic, intrinsically strongly coupled theories using conventional $\mathcal{N}=1$ Lagrangian techniques.