Conformal weights of RSOS lattice models and their fusion hierarchies

The finite-size corrections, central charges c and conformal weights Δ of L-state restricted solid-on-solid lattice models and their fusion hierarchies are calculated analytically. This is achieved by solving special functional equations, in the form of inversion identity hierarchies, satisfied by the commuting row transfer matrices at critically. The results are all obtained in terms of Rogers dilogarithms. The RSOS models exhibit two distinct critical regimes. For the regime III/IV critical line, we find c = [3p/(p + 2)][1 − 2(p + 2)/r(r − p)] where L = r − 1 is the number of heights and p = 1, 2, … is the fusion level. The conformal weights are given by a generalized Kac formula Δ = {[rt − (r − p)s]2 − p2}/ 4pr(r − p) + (s0 − 1)(p − s0 + 1)/ 2p(p + 2) where s = 1, 2, …, r − 1; t = 1, 2, …, r − p − 1; 1 ⩽ s0 ⩽ p + 1 and s0 − 1 = ±(t − s) mod 2p. For p = 1, 2, these models are described by the unitary minimal conformal series and the discrete superconformal series, respectively. For the regime I/II critical line, we obtain c = 2(N − 1)/(N + 2) and Δ = l(l + 2)/4(N + 2) − m2/4N for the conformal weights, independent of the fusion level p, where N = L − 1, l = 0, 1, …, N and m = −l, −l + 2, …, l − 2, l. In this critical regime the models are described by ZN parafermion theories.