Conformal Field Theory and Graphs *

The ADE graphs – the root diagrams of the simply laced simple Lie algebras appeared in the framework of the two-dimensional conformal field theories (CFT) based on the affine algebra sl(2) k in the classification of modular invariant partition functions [1] and in some related lattice models [2]. The modular invariants are labelled by the Coxeter number h = k + 2 while their diagonal terms, encoding the scalar field content of the theory, are enumerated by the Coxeter exponents. Graphs describing the spectrum of some invariants for the sl(n) theories were proposed in [3] (Z Z n − orbifolds) and in [4] (some cases of sl(3)). The aim of the reviewed work [5] is to further extend these results exploiting some deeper relations with structures in CFT. We first recall some basic facts. The WZNW genus one partition functions are sesquilinear forms on the affine algebra characters with integer coefficients N λ ¯ λ , Z (n+k) (q) = λ, ¯ λ N λ ¯ λ χ λ (q) χ ¯ λ (¯ q) , λ , ¯ λ ∈ P (k+n) ++ , N ρ ρ = 1. (1) Here P (k+n) ++ is set of integrable heighest weights of the affine algebra sl(n) k (shifted by ρ = (1, 1,. .. , 1)). The Type I theories are those for which there exists an algebra extending the chiral algebra and the partition function may be recast in a block-diagonal form