Supersymmetric gauge theories with decoupled operators and chiral ring stability

We propose a general way to complete supersymmetric theories with operators below the unitarity bound, adding gauge-singlet fields which enforce the decoupling of such operators. This makes it possible to perform all usual computations, and to compactify on a circle. We concentrate on a duality between an $\mathcal{N}=1$ $SU(2)$ gauge theory and the $\mathcal{N}=2$ $A_3$ Argyres-Douglas [1,2], mapping the moduli space and chiral ring of the completed $\mathcal{N}=1$ theory to those of the $A_3$ model. We reduce the completed gauge theory to $3d$, finding a $3d$ duality with $\mathcal{N}=4$ SQED with two flavors. The naive dimensional reduction is instead $\mathcal{N}=2$ SQED. Crucial is a concept of chiral ring stability, which modifies the superpotential and allows for a $3d$ emergent global symmetry.