The A-D-E classification of minimal andA 1 (1) conformal invariant theories

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.…”

“…For affine models characterized by the affine Kac -Moody algebra of type su(2) (chiral algebra), the classification of modular invariant partition functions is well known [7], and was shown to be in one-to-one correspondence with ADE Dynkin diagrams. More recently [29], it was shown that if the theory is associated with the Dynkin diagram G, its modular invariant partition function is given by Z 0 = χW 0,0 χ where χ is a vector of the complex vector space 3 C n and W 0,0 is the toric matrix associated with the origin of the Ocneanu graph of the diagram G. This characterization of partition functions uses only the (quantum) geometry of the diagram G and does not refer to the theory of affine algebras; in this approach, for instance, the fact that χ could be interpreted as a character of an affine Lie algebra is not used; in particular, modular invariance is implemented by finite dimensional matrices representing SL(2, Z).…”

“…We shall stick to the notation using level as a subscript. 7 Truncation is made by removing the parts of the diagram with level higher than k; what we obtain is a truncated Weyl chamber ("a Weyl alcove"). 8 Warning, in the SU (2) case, we have two notations for the same objects since the subindex of An refers usually to the number of vertices (the rank), but in this particular case, k = n − 1, so that A k=n−1 = An.…”

[Experimental nuclear physics]

“…The defining equation of the latter can be viewed as Landau-Ginzburg potentials for a two-dimensional (2, 2) supersymmetric field theory having the SCFF as its infrared limit. The partition functions [53] of the ADE superconformal models at level k, as well as their coupings, are explicitly known [54] and for tensor product models the LG potential is simply the sum of the corresponding LG potential terms. The central charge c = Y~'= 13ki/(ki + 2) is the sum of the central charges of the factor theories and has to be nine to cancel the conformal anomaly.…”

A common limit of super Liouville theory and minimal models

“…For unitary theories, the classification has only been achieved for central charge c < 1 resulting in the well known ADE series of minimal models [1,2]. One might hope that the problem still remains tractable for the limiting value c = 1.…”