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.