Holographic Roberge Weiss transitions II: Defect theories and the Sakai Sugimoto model

We extend the work of [1], including an imaginary chemical potential for quark number into the Sakai Sugimoto model and codimension k defect theories. The phase diagram of these models are a function of three parameters, the temperature, chemical potential and the asymptotic separation of the flavour branes, related to a mass for the quarks in the boundary theories. We compute the phase diagrams and the pressure due to the flavours of the theories as a function of these parameters and show that there are Roberge Weiss transitions in the high temperature phases, chiral symmetry restored for the Sakai Sugimoto model and deconfined for the defect models, while at low temperatures there are no Roberge Weiss transitions. In all the models we consider the transitions between low and high temperature phases are first order, hence the points where they meet the Roberge Weiss lines are triple points. The pressure for the defect theories scales in the way we expect from dimensional analysis while the Sakai Sugimoto model exhibits unusual scaling. We show that the models we consider are all analytic in mu(2) when mu(2) is small.