An introduction to conformal field theory (hep-th 9910156) by Gaberdiel M.R.

By Gaberdiel M.R.

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Next we consider the transformation of this amplitude under a rotation by 2π; this is implemented by the M¨obius transformation exp(2πiL0), Ω (∞) e2πiL0 µ(z)µ(0) = e− 2πi 4 Ω (∞) µ(e2πi z)µ(0) 1 = z 4 A + B log(z) + 2πiB , (247) where we have used that the transformation property of vertex operators (39) also holds for non-meromorphic fields. On the other hand, because of (246) we can rewrite Ω (∞) e2πiL0 µ(z)µ(0) 1 = z 4 Ω (∞) e2πiL0 ω(0) + log(z)Ω(0) . e. L0 Ω = 0, L0 ω = Ω. Thus we find that the scaling operator L0 is not diagonalisable, but that it acts as a Jordan block 0 1 0 0 (251) on the space spanned by Ω and ω.

However, this does not seem to be correct since the triplet algebra [150] has only finitely many irreducible representations, but contains indecomposable representations in their fusion products that lead to logarithmic correlation functions [151]. Logarithmic conformal field theories are not actually pathological; as was shown in [41] 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.

Conformal Field Theory 30 1 1 (22 + 5c) (m − n) (2m2 − mn + 2n2 − 8) Lm+n , (156) 48 30 where Λk are the modes of a quasiprimary field of conformal weight hΛ = 4. This field is a normal ordered product of L with itself, and its modes are explicitly given as + ∞ Λn = Ln−k Lk + k=−1 −2 k=−∞ Lk Ln−k − 3 (n + 2) (n + 3) Ln . 10 (157) One can check that this set of commutators satisfies the Jacobi-identity. ) Subsequently, various classes of W -algebras have been constructed [79–86]. There have also been attempts to construct systematically classes of W -algebras [87, 88] following [89].

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