Surface and corner free energies of the self-dual Potts model

We consider the bulk, vertical surface, horizontal surface and corner free energies $f_b, f_s, f'_s, f_c$ of the anisotropic self-dual $Q$-state Potts model for $Q > 4$. $f_b$ was calculated in 1973[1]. For $Q<4$, $f_s, f'_s$ were calculated in 1989[2]. Here we extend this last calculation to $Q>4$ and find agreement with the conjectures made in 2012 by Vernier and Jacobsen (VJ)[3] for the isotropic case. All these four free energies satisfy inversion and rotation relations. Together with some plausible analyticity assumptions, these provide a less rigorous, but much simpler, way of determining $f_b, f_s, f'_s$. They also imply that $f_c$ is independent of the anisotropy, being a function only of $Q$, in which respect they resemble the order parameters of the associated six-vertex model. Hence VJ's conjecture for $f_c$ should apply to the full anisotropic model.