Universal Planar Abelian Duals for 3d $\mathcal{N}=2$ Unitary CS-SQCD

Abstract

We provide an explicit planar Abelian dual for three-dimensional $\mathcal{N}=2$ $U(N)_k$ SQCD with $F$ fundamental chiral multiplets. This construction covers the entire $(N, F, k)$ parameter space (provided supersymmetry is unbroken), offering a unified framework for the infrared physics of these theories. Our results generalize a recently discovered class of chiral-planar dualities, which were previously limited to the locus $F = 2|k| + 2N$, which is a mass deformation of $\mathcal{N}=4$ mirror symmetry plus a restricted set of additional mass deformations. By developing a systematic algorithm to track the flow of the dual theory under generic mass deformations, we establish the planar Abelian quiver not merely as a specific dual description, but as a universal tool for analyzing 3d gauge dynamics.

Figures (13)

• Figure 1: Phase diagram of $U(N)_k$ SQCD with $F$ fundamental fields in the $(F,\,k)$-plane for fixed $N$. The diagram is partitioned into various zones, within which the planar Abelian duals take qualitatively distinct forms.
• Figure 2: The planar mirror dual of $U(3)_{-1/2}$ SQCD with $7$ fundamental chiral multiplets is an $\mathcal{N}=4$ descendant theory. Starting from this UV theory, we study the RG flow trajectories induced by $\pm m$ real mass deformations. In this figure, we display the first three levels of this deformation sequence. Specifically, we begin with an electric theory containing 7 fundamental chirals and follow the flow to theories with 4 fundamental chirals. We also highlight the location of these theories on the $(k,F)-$plane in the top-right corner.
• Figure 3: We continue analyzing the RG flow illustrated in Figure \ref{['fig: zoology']}. Starting from $U(3)_k$ with 4 fundamental multiplets, we study a single mass deformation down to 3 fundamental multiplets. Note that the dual of $U(3)_{-1/2}$ with 3 flavors is a trivial TQFT, while that of $U(3)_{-3/2}$ with 3 flavors is a free chiral. We highlight the location of these theories in the $(k,F)-$plane in the bottom-left corner.
• Figure 4: We continue the analysis of RG flow trajectories induced by $\pm m$ deformations, extending those considered in Figure \ref{['fig: zoo3']}. We focus on the RG flow trajectories of $U(3)_{-5/2}$ and $U(3)_{-3/2}$ SQCD with 3 fundamental chirals and follow these flows until all matter is integrated out. Notice that since the dual of $U(3)_{-3/2}$ with 3 fundamentals is a single free chiral, when we perform a mass deformation, we flow to a trivial theory. The theory with the $(\star)$ label is a gapped theory; in fact, the dual theory is simply a BF coupling. We highlight the location of these theories in the $(k,F)-$plane in the bottom-left corner.
• Figure 5: Examples of planar Abelian duals for $U(2)_k$ SQCD with matter content $[3,3]$, $[2,3]$, $[2,2]$, and $[1,3]$, for various values of $k$ are shown here. All dualities are obtained by systematically tracking real mass deformations in the planar Abelian theory, following the same strategy outlined in Section \ref{['sec: landscape']} for the case of $[F,0]$ flavors. A positive (negative) real mass for a fundamental chiral multiplet shifts the CS level by $+1/2$ ($-1/2$), while a positive (negative) real mass for an anti-fundamental shifts it by $-1/2$ ($+1/2$). The compact quiver notation is summarized in \ref{['eq:quiv:notation_compact']}. Further details of the theories, including the superpotential and R-charge assignments, follow the same rules described below \ref{['eq: Dual_k0']}.