The Riemann Surface of the Chiral Potts Model Free Energy Function

In a recent paper we derived the free energy or partition function of the N-state chiral Potts model by using the infinite lattice "inversion relation" method, together with a non-obvious extra symmetry. This gave us three recursion relations for the partition function per site T_pq of the infinite lattice, Here we use these recursion relations to obtain the full Riemann surface of T_pq. In terms of the t_p, t_q variables, it consists of an infinite number of Riemann sheets, each sheet corresponding to a point on a (2N-1)-dimensional lattice (for N>2). The function T_pq is meromorphic on this surface: We obtain the orders of all the zeros and poles. For N odd, we show that these orders are determined by the usual inversion and rotation relations (without the extra symmetry), together with a simple linearity ansatz. For N even, this method does not give the orders uniquely, but leaves only [(N+4)/4] parameters to be determined.