Twisted partition functions for ADE boundary conformal field theories and Ocneanu algebras of quantum symmetries

Abstract

For every ADE Dynkin diagram, we give a realization, in terms of usual fusion algebras (graph algebras), of the algebra of quantum symmetries described by the associated Ocneanu graph. We give explicitly, in each case, the list of the corresponding twisted partition functions.We dedicate this article to the memory of our friend Prof. Juan A. Mignaco deceased, 6 June 2001.

Torus structure on graphs and twisted partition functions for minimal and affine models

“…The non commutativity of Oc(D 4 ) does not show up in the fundamental twisted partition functions, indeed, although 2ǫ = ǫ2 in this algebra [11] (actually ǫ2 = 2 ′ ǫ), the toric matrices associated with these two points are the same. 25 Classical symmetries of the D4 diagram are described by the non commutative group algebra of the permutation group S3, this non commutativity also shows up at the quantum level in the structure of Oc(D4).…”

“…Vertices σ a of E 6 are labelled 0, 1, 2, 5, 4, 3 as shown on the picture. A 11 acts on E 6 , hence A 11 also acts from the left and from the right on the Ocneanu algebra 14 of quantum symmetries which can be shown to be equal ( [9], [11]) to Oc(E 6 ) = E 6 ⊗ A 3 E 6 . It has dimension 12.…”

“…From the diagram E 6 alone, we can determine the six essential matrices E a of dimension (11,6), as explained before. Rows of E 0 give the restriction (branching) rules A 11 → E 6 and columns give the induction rules.…”

“…These partition functions are the ones that involve only the combination of characters (1, 7), (4,8), (5,11), i.e., the adapted vector w of section 3.1.5. The symmetry relations for this family will read: Z Calling φ the characters vector of 110 components (only 55 distinct conformal weight), the 15 twisted partition functions involving the characters of type φ i1 + φ i7 , φ i4 + φ i8 , φ i5 + φ i11 take the following general form: …”

Quantum symmetries of face models and the double triangle algebra

“…Below we give the list of them together with the corresponding physical interpretation. In this respect it is worth remarking that we are interested in the set 6 of trigonometric face models associated to a graph G 6 We speak of a set because there are many trigonometric face models associated to the as in subsection 2.1.…”