Lattice models, cylinder partition functions, and the affine coxeter element

Abstract

The partition functions of the affine Pasquier models on the cylinder are calculated in the continuum limit. The partition functions of the models based upon the Â(_n) cycle graphs are first found from the appropriate Coulomb-gas equivalence. Their relationship with the D(_n) and Ề(_6,7,8) models is established by constructing an affine analogue to the classical intertwiners using a Temperley-Lieb algebraic equivalence. From this relationship, each of the partition functions is constructed. We write our results in terms of 'generating polynomials' establishing explicitly the precise operator content of the conformally invariant continuum field theories. A numerical study is undertaken to establish the validity of the partition functions as calculated. We conclude that the partition functions calculated are correct. The partition functions are further studied and the connection with the McKay correspondence established. We establish a simple form for the partition functions in terms of degenerate c = 1 Virasoro characters and Chebychev polynomials of the second kind. From this, we establish the role within the partition functions played by the affine Coxeter element, a particular member of the Weyl group associated with the defining graph of the model. Some of the resulting consequences of this role are explored.