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

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.

💡 Research Summary

This paper investigates both unitary and non‑unitary A‑D‑E Virasoro minimal models placed on a cylindrical geometry, with topological defect lines running along the non‑contractible cycle. The authors introduce a universal combinatorial object – the coset graph (A\otimes G/\mathbb Z_2) – constructed from the tensor product of two simply‑laced Dynkin diagrams (A) and (G) and then quotiented by the (\mathbb Z_2) graph automorphism. They demonstrate that this single graph simultaneously encodes four distinct pieces of data that are central to the boundary and defect conformal field theory (CFT) of these models:

1. Coset graph fusion algebra – The graph provides the adjacency matrices that serve as the fusion matrices for elementary defect lines (Verlinde‑type for the (A) sector and Ocneanu‑type for the (G) sector). These matrices form non‑negative integer matrix representations (nimreps) of the associative fusion algebra, reproducing the known fusion rules of the minimal models.

2. Affleck‑Ludwig boundary g‑factors – The largest eigenvalue of the graph’s adjacency matrix, (\lambda_{\max}=2\cos(\pi/m)) for unitary models (or (2\cos