Fused RSOS lattice models as higher-level nonunitary minimal cosets

We consider the Forrester–Baxter RSOS lattice models with crossing parameter λ = (m′−m)π/m′ in Regime III. In the continuum scaling limit, these models are described by the minimal models $M(m,m^{\prime })$. We conjecture that, for λ < π/n, the n × n fused RSOS models with $n⩾2$ are described by the higher-level coset ${\left(A_1^{\left(1\right)}\right)}_k\otimes {\left(A_1^{\left(1\right)}\right)}_n/{\left(A_1^{\left(1\right)}\right)}_{k+n}$ at fractional level k = nM/(M′−M)−2 with $(M,M^{\prime })=(nm-(n-1)m^{\prime },m^{\prime })$. To support this conjecture, we investigate the one-dimensional sums arising from Baxter's off-critical corner transfer matrices. In unitary cases (m = m′−1) it is known that, up to leading powers of q, these coincide with the branching functions $b_{r,s,\ell }^{m^{\prime }-n,m^{\prime },n}\left(q\right)$. For general nonunitary cases (m < m′−1), we identify the ground state one-dimensional RSOS paths and relate them to the quantum numbers (r, s, ℓ) in the various sectors. For n = 1, 2, 3, we obtain the local energy functions H(a, b, c) in a suitable gauge and verify that the associated one-dimensional sums produce finitized forms that converge, as N becomes large, to the fractional level branching functions $b_{r,s,\ell }^{M,M^{\prime },n}\left(q\right)$. Extending the work of Schilling, we also conjecture finitized bosonic branching functions $b_{r,s,\ell }^{M,M^{\prime },n;\left(N\right)}\left(q\right)$ for general n and check that these agree with the one-dimensional sums for n = 1, 2, 3 out to system sizes N = 14. Lastly, the finitized Kac characters ${\chi }_{r,s,\ell }^{P,P^{\prime },n;\left(N\right)}\left(q\right)$ of the n × n fused logarithmic minimal models $LM(p,p^{\prime })$ are obtained by taking the logarithmic limit $m,m^{\prime }\to \infty$ with m′/m→p′/p+. (paper)