Ocneanu Algebra of Seams: Critical Unitary $E_6$ RSOS Lattice Model

TL;DR

This work develops a lattice-based realization of the Ocneanu algebra of seams for critical A-D-E RSOS models, focusing on type I theories and the exceptional E6 example. By constructing integrable seams from commuting column transfer matrices, it shows that A-series seams obey Verlinde fusion while D/E seams realize graph fusion and the Ocneanu algebra, capturing the quantum symmetry of the transfer matrices. The E6 case is made explicit through extended toric matrices, a Hilbert-basis decomposition, and a quotient ring description that yields a 1D representation in terms of Virasoro x Virasoro quantum dimensions; the seam fusion reproduces the known toric and twisted partition functions. Overall, the lattice seam formalism provides a precise, computable bridge between integrable lattice models and the continuum CFT defects, with potential applications to higher-rank and nonunitary generalizations.

Abstract

We consider the $A$ series and exceptional $E_6$ Restricted Solid-On-Solid lattice models as prototypical examples of the critical Yang-Baxter integrable two-dimensional $A$-$D$-$E$ lattice models. We focus on type I theories which are characterized by the existence of an extended chiral symmetry in the continuum scaling limit. Starting with the commuting family of column transfer matrices on the torus, we build matrix representations of the Ocneanu graph fusion algebra as integrable seams for arbitrary finite-size lattices with the structure constants specified by Petkova and Zuber. This commutative seam algebra contains the Verlinde, fused adjacency and graph fusion algebras as subalgebras. Our matrix representation of the Ocneanu algebra encapsulates the quantum symmetry of the commuting family of transfer matrices. In the continuum scaling limit, the integrable seams realize the topological defects of the associated conformal field theory and the known toric matrices encode the twisted conformal partition functions.

Figures (5)

• Figure 1: Dynkin diagrams of the classical simply-laced $sl(2)$$A$-$D$-$E$ Lie algebras. The nodes associated with the identity and the fundamental are labelled 1 and 2 respectively. The fundamental is the unique neighbour of the identity. Also shown are the Coxeter numbers $g$, exponents ${\rm Exp}(G)$, the type I or II and the so-called parent graphs $H\ne G$. The diagram automorphism group $\Gamma$ is generated by a single $\mathbb{Z}_2$ automorphism $\sigma$. The $D_4$ graph is an exception having the noncommutative automorphism group $\Bbb S_3$. The eigenvalues of $G$ are $2\cos \tfrac{s\pi}{g}$ with $s\in {\rm Exp}(G)$. By abuse of notation, we use $G$ to denote the graph, the set of its vertices with cardinality $|G|$ and its adjacency matrix so that $2I-G$ is the Cartan matrix. The meaning of $G$ should be clear from context.
• Figure 2: An $N\times M$ periodic lattice with $(N,M)=(10,8)$ showing (i) the row transfer matrix $\hbox{$\mathbf T$}_h^x(u)$ with seam segment $x=(\kappa,a,b)$, (ii) the column/seam transfer matrices $\hbox{$\mathbf T$}(u)$, $\hbox{\boldmath $\mathbf n$}_{s}$, $\widehat{\mathbf N}_a$ and $\overline{\widehat{\mathbf N}}_b$ as explained in Section \ref{['sec:Ocneanu']} and (iii) the ${\mathbb Z}_2$ diagram automorphism seam $\boldsymbol\sigma^\kappa$. The composite seam is the matrix product $\hbox{\bf T}_x=\boldsymbol\sigma^\kappa\,\widehat{\mathbf N}_a\,\overline{\widehat{\mathbf N}}_b$. The seam $\hbox{\bf T}_x$ has the same twisted partition function as the seam $\widehat{\mathbf N}_a\,\overline{\widehat{\mathbf N}}_b\,\boldsymbol\sigma^\kappa$. We assume $r=1$ so there is no $r$-type seam shown. The labels $W_\kappa, W_a$ and $W_b$ indicate that special face weights are assigned to these faces.
• Figure 3: The $6\times 6$ array of the extended toric matrices $\widehat{P}_{a,b}$ of $E_6$. The 12 basis matrices (\ref{['basis']}) are linearly independent. The other 24 matrices are given by the quantum symmetry (\ref{['quant1']}).
• Figure 4: Ocneanu fusion graph $\widetilde{E}_6$ of $E_6$ with 12 vertices and two alternative labellings of nodes. The 12 vertices are in bijection with (i) the Hilbert basis $\widehat{P}_\eta$ (\ref{['basis']}) with $\eta=1,2,\ldots,12$ and (ii) the linearly independent integrable seams $\widehat{{\mathbf P}}_{\eta}$ with $\eta=1,2,\ldots,12$ as in (\ref{['hatPx']}). The red and blue lines (solid or dashed) show the action of the fundamental seams $\widehat{{\mathbf P}}_{2}=\widehat{{\mathbf P}}_{2,1}$ with $\widehat{{\mathbf P}}_{7}=\widehat{{\mathbf P}}_{\bar{2}}=\widehat{{\mathbf P}}_{1,2}$ respectively. This action interlocks four copies of the $E_6$ graph. The seams satisfy $\overline{\widehat{{\mathbf P}}}_{\eta}={\widehat{{\mathbf P}}}_{\bar{\eta}}$ where $\widehat{\mathbf P}_a=\widehat{\mathbf P}_{a,1}$ and $\widehat{\mathbf P}_{a+6}=\widehat{\mathbf P}_{a,2}$ with $a=1,2,\ldots,6$.
• Figure 5: Cayley table of the (commutative) $E_6$ Ocneanu algebra of seams with the compact notation $\eta=\widehat{\mathbf P}_{\eta}$ with $\eta=1,2,\ldots,12$. The fusion of these seams reproduces the known $E_6$ Ocneanu Cayley table CoqHuerta2003. The top-left quadrant is the Cayley table of the $E_6$ graph fusion algebra.