Supersymmetric gauge theories with decoupled operators and chiral ring stability

TL;DR

This work proposes a general method to complete supersymmetric gauge theories that harbor operators below the unitarity bound by coupling each such operator to a gauge-singlet flipping field β_O via W = β_O O, ensuring O vanishes in the chiral ring and enabling standard computations and circle compactifications. The authors refine the 4d duality between an N=1 SU(2) gauge theory and the A3 Argyres-Douglas theory by identifying an inconsistent superpotential term and enforcing a chiral ring stability condition that dictates dropping that term, yielding a consistent stable Lagrangian whose moduli space matches A3 AD. They demonstrate that compactification to 3d produces an emergent N=4 Abelian dual (U(1) with two flavors) and show complete mapping of chiral generators and partition functions, supported by S^3 calculations and duality checks. The paper further shows that preserving the flipping-field structure is essential for a correct 3d descent and SUSY enhancement, as naive reductions fail to reproduce the expected dualities. Overall, the results provide a robust framework for handling decoupled operators in SUSY gauge theories and clarifying the 4d–3d correspondence via chiral ring stability and Abelianization.

Abstract

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.