Some comments on the solvable chiral potts model

We present a simple proof of the conjecture produced by Baxter, Perk and Au-Yang on the structure of the normalization factorR(p, q, r) corresponding to their new solution of the star-triangle equation related with the generalized Fermat curve. Some important properties of the underlying curvex ^N y ^N+x ^N+y ^N+1/k ²=0 for theN=3 state case are also established. Particularly, we calculate exactly its matrix of theb-periods for some normalized basis of holomorphic differentials. We also show that associated four-dimensional theta function may be decomposed into a sum containing 12 terms, each term being the product of four one-dimensional theta functions. We also derive Picard-Fuchs equations for the periods of holomorphic differentials of the same curve. The remarkable appearance of the hypergeometric functions in our answers seems to be closely related with an expression for the groundstate energy per site, obtained for the superintegrable case by Albertini, Perk, and McCoy and independently by Baxter, although for a moment the connection is not clear.