On the classification of diagonal coset modular invariants

Abstract

We relate in a novel way the modular matrices of GKO diagonal cosets without fixed points to those of WZNW tensor products. Using this we classify all modular invariant partition functions of su(3) k ⊕su(3) 1 /su(3) k+1 for all positive integer level k, and su(2) k ⊕ su(2) ℓ /su(2) k+ℓ for all k and infinitely many ℓ (in fact, for each k a positive density of ℓ). Of all these classifications, only that for su(2) k ⊕ su(2) 1 /su(2) k+1 had been known. Our lists include many new invariants. *

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

“…A classification of modular invariant partition functions for conformal field theories of SU (2) type was obtained at the same time, i.e., at the end of the eighties, by [7] in a celebrated paper. Later, T. Gannon (and collaborators) could obtain ( [21]) similar results for conformal field theories based on more general affine Kac -Moody algebras.…”

“…but if our only purpose is to determine Oc(G), it is simpler to find a short cut. One possibility is to use the fact that we already know, in many cases, the expression of the modular invariant (as calculated by [7] for SU (2) and [21] for SU (3)); such a technique was apparently followed by A. Ocneanu himself in his determination of the irreducible quantum symmetries x, also called "irreducible connections", associated with a given diagram. However, if we do not want to use this a priori knowledge, there is another technique, which uses modular properties of the diagram; this was one of the purposes of the article [12].…”

“…The structure of the corresponding partition function Z 21,21 = χ.W 21,21 .χ is better understood if we use this last expression, leading to…”

“…than if we just consider its fully developed form obtained by using the explicit expression for matrix W 21,21 . The same toric matrix can also be obtained, using our "second algorithm", as a linear combination of toric matrices with a single twist; indeed, the Oc(E 6 ) multiplication table tells us that…”

“…The dimension of the space of paths on E 5 is infinite, but when we restrict our attention to essential paths (one type of essential 21 We write "modular" but the relevant group is SL(2, Z), not P SL(2, Z). path for every vertex of A 5 ), one finds 21 possibilities i.e., 21 blocks of dimensions (d j , d j ) for the first algebra structure of BG.…”