Fused RSOS Lattice Models as Higher-Level Nonunitary Minimal Cosets

We consider the Forrester-Baxter RSOS lattice models with crossing parameter $\lambda =(m^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}m)\pi \mathrm{/}{m}^{\mathrm{\prime }}$ in Regime~III. In the continuum scaling limit, these models are described by the minimal models $\mathcal{M}(m,m^{\mathrm{\prime }})$. We conjecture that, for $\lambda <\pi \mathrm{/}n$, the $n\times n$ fused RSOS models with $n\ge 2$ are described by the higher-level coset $(A^{(1)}_1)_k\otimes (A^{(1)}_1)_n/(A^{(1)}1){k+n}$atfractionallevel$k=nM/(M’!-!M)-2$with$(M,M’)=\big(nm-(n!-!1)m’,m’\big)$. To support this conjecture, we investigate the one-dimensional sums arising from Baxter’s off-critical corner transfer matrices. In unitary cases ($m=m^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}1$) it is known that, up to leading powers of $q$, these coincide with the branching functions $b_{r,s,\mathrm{\ell }}^{m^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}-n,m^{\mathrm{\prime }}\text{\hspace{0.17em}⁣},n}(q)$. For general nonunitary cases ($m<m^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}1$), we identify the ground state one-dimensional RSOS paths and relate them to the quantum numbers $(r,s,\mathrm{\ell })$ 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 $b_{r,s,\mathrm{\ell }}^{M,M^{\mathrm{\prime }}\text{\hspace{0.17em}⁣},n}(q)$. Extending the work of Schilling, we also conjecture finitized bosonic branching functions $b_{r,s,\mathrm{\ell }}^{M,M^{\mathrm{\prime }}\text{\hspace{0.17em}⁣},n;(N)}(q)$ 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 ${\chi }_{r,s,\mathrm{\ell }}^{P,P^{\mathrm{\prime }}\text{\hspace{0.17em}⁣},n;(N)}(q)$ of the $n\times n$ fused logarithmic minimal models $\mathcal{L}\mathcal{M}(p,p^{\mathrm{\prime }})$ are obtained by taking the {\em logarithmic limit/} $m,m^{\mathrm{\prime }}\to \mathrm{\infty }$ with $m\mathrm{/}{m}^{\mathrm{\prime }}\to p\mathrm{/}{p}^{\mathrm{\prime }}+$.