The many faces of Ocneanu cells

By V. Petkova, Jean-Bernard Zuber. 2001

139 citations recorded by the source corpus.

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 $\tilde A$ 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 $\tilde A$. 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.

Subfields: Atomic and Molecular Physics, and Optics, Discrete Mathematics and Combinatorics, Geometry and Topology

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In Atomic and Molecular Physics, and Optics

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Citations in the corpus

TFT construction of RCFT correlators: III: simple currents (2004, background): “(i) The fusing matrices G[A] (or rather, their inverses F[A]) also appear in an approach to CFT based on graphs and cells [22, 23 ], where the notation (1)F is used.”
TFT construction of RCFT correlators I: Partition functions (2002, result): “We suggest that our results can be interpreted in terms of non-commutative geometry over the modular tensor category of Moore--Seiberg data.”
Generalised twisted partition functions (2000, background): “As will be explained below, all Ṽx y may be reconstructed from the simpler Ocneanu matrices Ṽ1 x.”
TFT construction of RCFT correlators IV: Structure constants and correlation functions (2004, background): “2 : In view of the results in [14, 22, 46, 4, 23] one might have expected that the F-matrices appear in (4.”
Module categories, weak Hopf algebras and modular invariants (2001, methodology): “(ii) Some version of Theorem 4.1 is known in the theory of operator algebras and in physics, see e.g., [O1, BEK, PZ ].”
The virtue of defects in 4D gauge theories and 2D CFTs (2010, background): “[9, 10, 11]) in a 2d CFT are operators supported on closed curves characterized by the property that they commute with both the holomorphic and antiholomorphic energy-momentum tensor of the CFT.”