Domain walls in high-T SU(N) super Yang-Mills theory and QCD(adj)

We study the domain walls in hot $4$-D $SU(N)$ super Yang-Mills theory and QCD(adj), with $n_f$ Weyl flavors. We find that the $k$-wall worldvolume theory is $2$-D QCD with gauge group $SU(N-k)times SU(k) times U(1)$ and Dirac fermions charged under $U(1)$ and transforming in the bi-fundamental representation of the nonabelian factors. We show that the DW theory has a $1$-form $mathbb Z_{N}^{(1)}$ center symmetry and a $0$-form $mathbb Z_{2Nn_f}^{dchi}$ discrete chiral symmetry, with a mixed t Hooft anomaly consistent with bulk/wall anomaly inflow. We argue that $mathbb Z_{N}^{(1)}$ is broken on the wall, and hence, Wilson loops obey the perimeter law. The breaking of the worldvolume center symmetry implies that bulk $p$-strings can end on the wall, a phenomenon first discovered using string-theoretic constructions. We invoke $2$-D bosonization and gauged Wess-Zumino-Witten models to suggest that $mathbb Z_{2Nn_f}^{dchi}$ is also broken in the IR, which implies that the $0$-form/$1$-form mixed t Hooft anomaly in the gapped $k$-wall theory is saturated by a topological quantum field theory. We also find interesting parallels between the physics of high-temperature domain walls studied here and domain walls between chiral symmetry breaking vacua in the zero temperature phase of the theory (studied earlier in the semiclassically calculable small spatial circle regime), arising from the similar mode of saturation of the relevant t Hooft anomalies.