Star-triangle equations and some properties of algebraic curves connected with the integrable chiral Potts model (Q915939)

The authors obtain the following results concerning the integrable model of Potts. Firstly, they calculate the explicite form of the factor R(p,q,r) appearing in the startriangle equation. The second result is the explicit expression for the elements of matrix B-periods of the curve \[ x^ Ny^ N+x^ N+y^ N+1/k^ 2=0,\text{ for } N=3. \] The third result is the equation of Piccard-Fuchs for the above curve. In general, if an algebraic curve of genus g depending upon some parameter, every integral along an arbitrary closed contour of holomorphic differential satisfies, as functions of this parameter, some ordinary linear differential equation of order 2g (equation of Piccard and Fuchs). The authors prove that for the above curve with \(N=3\), the equation can be strongly simplified and reduced to hypergeometric equations of special form, such as \[ a(1-a)\ddot z_ 1+(1-2a)\dot z_ 1-(2/9)z_ 1=0, \] where \(\dot z_ 1=dz_ 1/da\).