On the Equivalent Theory of the Generalized \tau^{(2)}-model and the Chiral Potts Model with two Alternating Vertical Rapidities

By the Baxter’s $Q_{72}$-operator method, we demonstrate the equivalent theory between the generalized ${\tau }^{(2)}$-model (other than two special cases with a pseudovacuum state) and the $N$-state chiral Potts model with two alternating vertical rapidities, where the degenerate models are included. As a consequence, the theory of the XXZ chain model associated to cyclic representations (with the parameter $\varsigma$) of $U_{\mathsf{q}}(s{l}_2)$ with ${\mathsf{q}}^N=1$ for odd $N$ is identified with either (for ${\varsigma }^N=1$) the chiral Potts model with two superintegrable vertical rapidities, or (for ${\varsigma }^N\mathrm{\ne }1$) the degenerate model for the selfdual solution of the star-triangle relation. In all these identifications, the transfer matrices $T, \hat{T}$ofthechiralPottsmodel(includingthedegenerateones)serveasthe$Q_R, Q_L$-operatorsofthecorresponding$\tau^{(2)}$-model, so that the functional relations hold as in the solvable $N$-state chiral Potts model.