Domain Walls in 4d N=1 Supersymmetric Yang-Mills

Understand

$4d$ ${\mathcal N}=1$ super Yang-Mills (SYM) with simply connected gauge group $G$ has $h$ gapped vacua arising from the spontaneously broken discrete $R$-symmetry, where $h$ is the dual Coxeter number of $G$.

• Therefore, the theory admits stable domain walls interpolating between any two vacua, but it is a nonperturbative problem to determine the low energy theory on the domain wall.
• We put forward an explicit answer to this question for all the domain walls for $G=SU(N),Sp(N), Spin(N)$ and $G_2$, and for the minimal domain wall connecting neighboring vacua for arbitrary $G$.
• We propose that the domain wall theories support specific nontrivial topological quantum field theories (TQFTs), which include the Chern-Simons theory proposed long ago by Acharya-Vafa for $SU(N)$.