Solvable Models in Statistical Mechanics and Riemann Surfaces of Genus Greater than One

Abstract

This chapter discusses the solvable models in statistical mechanics and Riemann surfaces of genus greater than one. Most recently, it was discovered that there is an N-state generalization of the Ising model which seems to possess all of its nice properties. This model is the chiral Potts model or ZN symmetric model. A most natural place to commence the investigation of any two-dimensional statistical mechanical model is to study its transfer matrix. The Ising model (N=2) holds its distinguished place amongst solvable models because of its relation to a free fermion field theory. This relation is what is ultimately behind all the various nonlinear difference equations, nonlinear differential equations, and deformation theory results which have been derived for the Ising model. If the spins of the Ising model are not on the diagonal, the correlations involve elliptic integrals of the third kind, and this has an immediate extension to Abelian integrals of the third kind for arbitrary Riemann surfaces.