The Infrared Physics of Bad Theories

TL;DR

This work provides a comprehensive algebraic analysis of the moduli spaces of 3d ${\rm N}=4$ ${U(N)}$ SQCD with ${N_f}$ fundamentals, revealing how IR physics in any vacuum emerges from local Coulomb-Higgs geometry. By classifying good, ugly, and bad theories through the geometry of the Coulomb branch, it shows that bad theories flow to the IR fixed points of good theories plus free twisted hypermultiplets, and it disproves a global Seiberg-like duality for the bad range ${N_f}\le 2N-2$. A distinguished symmetric vacuum ${\mathcal P}$ captures a local IR equivalence to the proposed dual, and the authors connect this to sphere partition functions and the UV/IR ${R}$-symmetry mixing. The results clarify vacuum-dependent IR dynamics, provide a precise embedding of good moduli into bad moduli, and offer a framework to extend to other gauge groups and deformations, with implications for dualities and holographic interpretations in 3d ${\rm N}=4$ theories.

Abstract

We study the complete moduli space of vacua of 3d $\mathcal{N}=4$ $U(N)$ SQCD theories with $N_f$ fundamentals, building on the algebraic description of the Coulomb branch, and deduce the low energy physics in any vacuum from the local geometry of the moduli space. We confirm previous claims for good and ugly SQCD theories, and show that bad theories flow to the same interacting fixed points as good theories with additional free twisted hypermultiplets. A Seiberg-like duality proposed for bad theories with $N \le N_f \le 2N-2$ is ruled out: the spaces of vacua of the putative dual theories are different. However such bad theories have a distinguished vacuum, which preserves all the global symmetries, whose infrared physics is that of the proposed dual. We finally explain previous results on sphere partition functions and elucidate the relation between the UV and IR $R$-symmetry in this symmetric vacuum.

Figures (3)

• Figure 1: A schematic picture of the Coulomb branch of the bad theory $\mathcal{C}_{\rm bad}$ with its nested sequence of singular subloci $\mathcal{C}_1 \supset \mathcal{C}_2 \supset \cdots \supset \mathcal{C}^\ast$ of increasing codimension. The Coulomb branch of the good theory $\mathcal{C}_{\rm good}$ is included into $\mathcal{C}_{\rm bad}$ as a codimension $2N-N_f$ subvariety, and its most singular point $\mathcal{P}$ lies on a non-maximal singular subvariety of $\mathcal{C}_{\rm bad}$.
• Figure 2: A schematic picture of the full moduli space of vacua, consisting of a Coulomb branch (blue), a Higgs branch (red) and a mixed branch (purple).
• Figure 3: Two mixed branches $\mathcal{C}_{N-r}\times \mathcal{H}_r$ (black) and $\mathcal{C}_{N-r-1}\times \mathcal{H}_{r+1}$ (red) intersecting on the common subvariety $\mathcal{C}_{N-r-1}\times \mathcal{H}_r$.