High-temperature domain walls of QCD with imaginary chemical potentials

We study QCD with massless quarks on ${\mathbb{R}}^3\times S^1$ under symmetry-twisted boundary conditions with small compactification radius, i.e. at high temperatures. Under suitable boundary conditions, the theory acquires a part of the center symmetry and it is spontaneously broken at high temperatures. We show that these vacua at high temperatures can be regarded as different symmetry-protected topological orders, and the domain walls between them support nontrivial massless gauge theories as a consequence of anomaly-inflow mechanism. At sufficiently high temperatures, we can perform the semiclassical analysis to obtain the domain-wall theory, and $2$d $U(N_{\mathrm{c}}-1)$ gauge theories with massless fermions match the 't~Hooft anomaly. We perform these analysis for the high-temperature domain wall of ${\mathbb{Z}}_{N_{\mathrm{c}}}$-QCD and also of Roberge-Weiss phase transitions.