Unitary and Nonunitary A-D-E minimal models: Coset graph fusion algebras, defects, entropies, SREEs and dilogarithm identities

TL;DR

The work shows that universal coset graphs $A\otimes G/\mathbb{Z}_2$ faithfully encode the fusion, boundary/defect $g$-factors, and relative entanglement structure of unitary and nonunitary $A$-$D$-$E$ minimal models. It builds a detailed bridge between continuum CFT data and lattice RSOS realizations via Yang–Baxter integrability, deriving T- and Y-system relations whose braid/bulk limits reproduce the coset fusion algebra and quantum-dimension data. Central charges and conformal weights emerge from dilogarithm identities evaluated on braid and bulk limits, using the generalized quantum dimensions $d^m_{j,r}$ and $d^{m'}_{j,s}$. The framework yields explicit prototypical models (Ising, Potts, Lee–Yang, etc.) with complete defect, boundary, and modular data, and shows that many CFT structures already appear at the lattice level, enabling a lattice-to-CFT dictionary for defects and SREEs.

Abstract

We consider both unitary and nonunitary A-D-E minimal models on the cylinder with topological defects along the non-contractible cycle of the cylinder. We define the coset graph $A \otimes G/\mathbb{Z}_2$ and argue that it encodes not only the (i) coset graph fusion algebra, but also (ii) the Affleck-Ludwig boundary g-factors; (iii) the defect g-factors (quantum dimensions) and (iv) the relative symmetry resolved entanglement entropy. By studying A-D-E restricted solid-on-solid models, we find that these boundary conformal field theory structures are also present on the lattice: defects (seams) are implemented by face weights with special values of the spectral parameter. Integrability allows the study of lattice transfer matrix T- and Y-system functional equations to reproduce the fusion algebra of defect lines. The effective central charges and conformal weights are expressed in terms of dilogarithms of the braid and bulk asymptotics of the Y-system expressed in terms of the quantum dimensions.

Figures (16)

• Figure 1: Dynkin diagrams of the classical simply-laced $A$-$D$-$E$ Lie algebras. The nodes associated with the identity and the fundamental are labelled by 1 and 2 respectively. We note that the fundamental is the unique neighbour of the identity. Also shown are the Coxeter numbers $m$, exponents ${\rm Exp}(G)$, the type I or II, the so-called parent $A$-$D$-$E$ graph $H\ne G$ and the diagram automorphism group $\Gamma$. It is the exponents of the parent graph $H$ that appear in the modular invariant partition functions as in Table \ref{['MIPFs']}. The $D_4$ graph is exceptional having the noncommutative automorphism group $\Bbb S_3$. The eigenvalues of $G$ are $2\cos \tfrac{\ell\pi}{m}$ with $\ell\in {\rm Exp}(G)$. By abuse of notation, we use $G$ to denote the graph, its adjacency matrix and its set of vertices with cardinality $|G|$ but the meaning should be clear from context.
• Figure 2: The conformal cylinder partition functions (\ref{['CylPFs']}) are generated by propagating the defect lines ${\cal L}_\mu$ (with $\mu=(r,a)$ or $\mu=(r,s)$) glued to the two boundaries to the center and fusing them. If $G$ is of $A$-type, then $n_{sb}{}^c$ reduces to the Verlinde structure constants $N_{ss'}{}^{s'\!'}$ and the cylinder partition functions are then compatible with the Kac symmetry since ${\cal L}_{(r,s)}={\cal L}_{(m-r,m'\!-s)}$. Similar arguments apply for more general topological defects.
• Figure 3: Vanden Broeck-Schwartz vBS extrapolations of degeneracy sequences for $c_\text{eff}$ (top row) and 2-boundary $\mathsf g$-factors (bottom row). The top row relates to (i) the central charge $c=c_\text{eff}=\tfrac{7}{10}$ of the tricritical Ising model ${\cal M}(4,5)$, (ii) the effective central charge $c_\text{eff}=\tfrac{2}{5}$ of the Lee-Yang model ${\cal M}(2,5)$. The bottom row relates to (iii-iv) the 2-boundary $\mathsf g$-factors $\tilde{\mathsf g}_{(1,1)|(1,1)}=0.525731\ldots$ and $\tilde{\mathsf g}_{(1,1)|(1,2)}=0.850651\ldots$ of the Lee-Yang model ${\cal M}(2,5)$. The values of $n$ in the sequences (\ref{['sequences']}) range from $n=1000$ to $n=11000$ in increments of $\Delta n=1000$. The convergence of the $\mathsf g$-factors is faster than the convergence of the effective central charges.
• Figure 4: The bipartite coset graph $\tilde{G}=A_2\otimes A_3/\mathbb{Z}_2=A_3$ and Kac tables of conformal weights and 2-boundary $\mathsf g$-factors $\mathsf g_{(1,1)|(r,s)}$ for the critical Ising model ${\cal M}(3,4)$ with $c=\tfrac{1}{2}$. Under the Kac table symmetry $(1,2)\equiv (2,2)$, so the nodes $(r,s)=(1,1),(2,2),(1,3)$ are simply labelled by $s=1,2,3$.
• Figure 5: The bipartite coset graph $\tilde{G}=\tilde{N}_2=A_3\otimes A_4/\mathbb{Z}_2$ and Kac tables of conformal weights and 2-boundary $\mathsf g$-factors $\mathsf g_{(1,1)|(r,s)}$ for the tricritical Ising model ${\cal M}(4,5)$ with $c=\tfrac{7}{10}$. Choosing the $r\!+\!s$ even sublattice, the nodes $(r,s)\in\mathbb{K}=\{(1,1),(2,2),(3,3),(2,4),(1,3),(3,1)\}$ are labelled by $\mu=1,2,\ldots,6$. The fundamental $(2,2)$ is labelled by $\mu=2$.