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

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We consider the bulk, vertical surface, horizontal surface and corner free energies fb, fₛ, f'ₛ, fc of the anisotropic self-dual Q-state Potts model for Q > 4. fb was calculated in 1973[1]. For Q<4, fₛ, 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 fb, fₛ, f'ₛ. They also imply that fc 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 fc should apply to the full anisotropic model.