By Gaberdiel M.R.
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Additional resources for An introduction to conformal field theory (hep-th 9910156)
If the number of irreducible representations is infinite. However, this does not seem to be correct since the triplet algebra  has only finitely many irreducible representations, but contains indecomposable representations in their fusion products that lead to logarithmic correlation functions . Logarithmic conformal field theories are not actually pathological; as was shown in  a consistent local conformal field theory that satisfies all conditions of a local theory (including modular invariance of the partition function) can be associated to this triplet algebra.
We therefore define, more precisely, the fusion rule Nijk to be the multiplicity with which the representation conjugate to φk appears in φi (u1 )φj (u2). The action of the meromorphic fields (or rather their modes) on the product of the two fields can actually be described rather explicitly using the comultiplication formula [40, 125, 126]:† let us denote by A the algebra of modes of the meromorphic fields. A comultiplication is a homomorphism ∆ : A → A ⊗ A, Vn (ψ) → ∆(1)(Vn (ψ)) ⊗ ∆(2)(Vn (ψ)) , (224) and it defines an action on the product of two fields as Vn (ψ) φi (u1 )φj (u2) = ∆(1)(Vn (ψ))φi (u1) ∆(2)(Vn (ψ))φj (u2) .
The equation that comes from the coefficient of xs−1 , s(s − 1) + s = s2 = 0 . (241) Generically, the indicial equation has two distinct roots (that do not differ by an integer), and for each solution of the indicial equation there is a solution of the original differential equation that is of the form (240). However, if the two roots coincide (as in our case), only one solution of the differential equation is of the form (240), and the general solution to (239) is F (x) = AG(x) + B [G(x) log(x) + H(x)] , (242) where G and H are regular at x = 0 (since s = 0 solves (241)), and A and B are constants.
An introduction to conformal field theory (hep-th 9910156) by Gaberdiel M.R.