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

TL;DR

This paper constructs matrix representations of the Ocneanu algebra for critical $E_6$ RSOS lattice models, linking lattice integrable seams to conformal field theory topological defects and quantum symmetries.

Contribution

It introduces a novel matrix representation of the Ocneanu algebra for $E_6$ RSOS models, connecting lattice models with conformal field theory defects.

Findings

Matrix representations encode quantum symmetries.

Seams realize topological defects in CFT.

Toric matrices encode twisted partition functions.

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.