Chiral Potts Rapidity Curve Descended from Six-vertex Model and Symmetry Group of Rapidities

TL;DR

This paper systematically derives the rapidity curve of the chiral Potts model from the six-vertex model, analyzing its symmetry group and functional equations, revealing eigenvalue degeneracies in superintegrable cases.

Contribution

It provides a detailed derivation of the rapidity curve and symmetry group of the chiral Potts model from the six-vertex model using fusion relations and functional equations.

Findings

Identification of the high genus rapidity curve

Symmetry group structure of the rapidity curve

Eigenvalue degeneracy in superintegrable case

Abstract

In this paper, we present a systematical account of the descending procedure from six-vertex model to the $N$-state chiral Potts model through fusion relations of $\tau^{(j)}$-operators, following the works of Bazhanov-Stroganov and Baxter-Bazhanov-Perk. A careful analysis of the descending process leads to appearance of the high genus curve as rapidities' constraint for the chiral Potts models. Full symmetries of the rapidity curve are identified, so is its symmetry group structure. By normalized transfer matrices of the chiral Potts model, the $\tau^{(2)}T$ relation can be reduced to functional equations over a hyperelliptic curves associated to rapidities, by which the degeneracy of $\tau^{(2)}$-eigenvalues is revealed in the case of superintegrable chiral Potts model.