The many faces of Ocneanu cells

TL;DR

This paper introduces a new chiral approach using Ocneanu double triangle algebra to unify algebraic structures in rational 2D conformal field theories and their lattice models, revealing new insights into boundary states, partition functions, and operator algebras.

Contribution

It defines covariant chiral vertex operators under the Ocneanu algebra, providing a unified algebraic framework for conformal field theories and lattice models.

Findings

Ocneanu cells are identified as 3j-symbols of a weak Hopf algebra.

Graphs G and e G encode boundary states and twisted partition functions.

Ocneanu quantum symmetry informs the operator algebra structure.

Abstract

We define generalised chiral vertex operators covariant under the Ocneanu ``double triangle algebra'' {\cal A}, a novel quantum symmetry intrinsic to a given rational 2-d conformal field theory. This provides a chiral approach, which, unlike the conventional one, makes explicit various algebraic structures encountered previously in the study of these theories and of the associated critical lattice models, and thus allows their unified treatment. The triangular Ocneanu cells, the 3j-symbols of the weak Hopf algebra {\cal A}, reappear in several guises. With {\cal A} and its dual algebra {hat A} one associates a pair of graphs, G and {\tilde G}. While G are known to encode complete sets of conformal boundary states, the Ocneanu graphs {\tilde G} classify twisted torus partition functions. The fusion algebra of the twist operators provides the data determining {\hat A}. The study of bulk field correlators in the presence of twists reveals that the Ocneanu graph quantum symmetry gives also an information on the field operator algebra.