Living on the walls of super-QCD

TL;DR

This work provides a comprehensive 3d worldvolume description for BPS domain walls in 4d ${\\mathcal{N}=1}$ SQCD with $F<N$, proposing a CS-matter theory ${U(k)_{N-\frac{k+F}{2},N-\frac{F}{2}}}$ coupled to $F$ fundamental flavors and an adjoint+singlet, with a mass parameter $m$ and two quartic interactions. The authors demonstrate a precise match between the 3d vacua and the 4d BPS wall solutions in the small-mass limit, and show a second-order wall-phase transition as $m_{4d}$ varies, including instances of supersymmetry enhancement at special parameter values. The framework reproduces known results in the large-mass limit (Acharya–Vafa) and extends them to the full $F<N$ regime, while also generalizing to ${oldsymbol{Sp(N)}}$ SQCD with $F<N+1$. A detailed analysis of symmetry-preserving and symmetry-breaking walls is provided, including algebraic reductions on mesonic space, a covering-space treatment of multi-valued superpotentials, and explicit examples for low ranks, yielding a coherent global picture of BPS domain walls in massive SQCD. Overall, the paper connects 4d wall dynamics to rich 3d CS-matter phases, revealing phase transitions, dualities, and possible SUSY enhancements that deepen the understanding of domain walls in supersymmetric gauge theories.

Abstract

We study BPS domain walls in four-dimensional $\mathcal{N}=1$ massive SQCD with gauge group $SU(N)$ and $F<N$ flavors. We propose a class of three-dimensional Chern-Simons-matter theories to describe the effective dynamics on the walls. Our proposal passes several checks, including the exact matching between its vacua and the solutions to the four-dimensional BPS domain wall equations, that we solve in the small mass regime. As the flavor mass is varied, domain walls undergo a second-order phase transition, where multiple vacua coalesce into a single one. For special values of the parameters, the phase transition exhibits supersymmetry enhancement. Our proposal includes and extends previous results in the literature, providing a complete picture of BPS domain walls for $F<N$ massive SQCD. A similar picture holds also for SQCD with gauge group $Sp(N)$ and $F < N+1$ flavors.

Figures (6)

• Figure 1: A qualitative picture of the phase diagram of $k$-walls of $SU(N)$ SQCD with $F<N$ flavors. In the small mass regime the $k$-wall theory has multiple vacua, parametrized by an integer $J$. In each vacuum, a topological theory is accompanied by a supersymmetric NLSM with target the complex Grassmannian $\mathrm{Gr}(J,F) = U(F) / \bigl[ U(J) \times U(F-J) \bigr]$. A similar phase diagram holds for $Sp(N)$ SQCD with $F<N+1$ flavors.
• Figure 2: Symmetry preserving $k$-walls in massive $SU(N)$ SQCD with $F<N$ flavors for $N=2,3$. Each figure refers to fixed values of $N,F$ and soliton sector $k$. We draw the $N$ vacua (red dots and circles) on the $\widetilde{M}$-plane, as well as the pre-image of a straight line in the $W$-plane connecting $W_{-\infty}$ to $W_{+\infty}$. The pre-image consists of $N$ curves, drawn in different colors. Paths connecting the vacua and thus corresponding to domain walls are solid lines, while the rest of the pre-image is dashed. If a path involves a jump from one sheet to another (namely, from one portion of the pre-image to another), we indicate the value of $\Delta$.
• Figure 3: Symmetry preserving $k$-walls in massive $SU(4)$ SQCD with $F=1,2,3$ flavors. Conventions are the same as in Figure \ref{['fig: SU(2) SU(3) DWs']}.
• Figure 4: Symmetry preserving $k$-walls in massive $SU(5)$ SQCD with $F$ flavors, for some selected values of $F$ and soliton sector $k$. Conventions are again as in Figure \ref{['fig: SU(2) SU(3) DWs']}.
• Figure 5: Symmetry breaking $k$-walls in massive $SU(N)$ SQCD with $N=3,4$ and $F=N-1$ flavors. Each figure refers to a specific domain wall for given values of $N,F,k,J$. We draw the (smooth) orbits in the complex plane of a solution to the differential equations (\ref{['polar DW ODEs']}), in which $n_+$ eigenvalues are equal to $\lambda_+$ and $n_-$ are equal to $\lambda_-$ ($n_+ + n_-=F$).