Algebraic geometry of the three-state chiral Potts model (Q5951515)

The chiral Potts model of statistical mechanics has been studied by a number of mathematical physicists and it has been formulated and solved in one and two-dimensions. In this paper, the authors use the tools of modern algebraic geometry to study the chiral Potts model with three-states. In this case, there appears the curve \(X\equiv x^3+y^3=k(1+x^3y^3)\) of genus 4 where \(k\notin\{0,1,-1\}\). The authors prove that \(\text{Aut}(X)=S_3\times S_3\) and the Jacobian \(J(X)\) is isogenous to the product of four elliptic curves which occur as pairs. They compute the degree of this isogeny and they calculate the direct image of theta divisor by this isogeny. This computation allows to expand the theta function of \(X\) as a sum of products of four elliptic theta functions and to show that the formula of \textit{V. B. Matveev} and \textit{A. O. Smirnov} [Lett. Math. Phys. 19, 179-185 (1990; Zbl 0715.14024)] is one of an infinite family of such identities. The chiral Potts model is not solved in this paper, however the authors present a systematic approach to the underlying mathematical aspects of the problem.