By Gaberdiel M.R.
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In order to prove this it is sufficient to consider the case where ψ and χ are both eigenvectors of L0 with eigenvalues hψ and hχ , respectively. Then we have dζ dw [VL (ψ), VR (χ)] = (ζ + 1)hψ (w + 1)hχ −1 V (ψ, ζ)V (χ, w) ζ w |ζ|>|w| dζ dw − (w + 1)hχ −1 (ζ + 1)hψ V (χ, w)V (ψ, ζ) ζ |w|>|ζ| w dζ dw = (ζ + 1)hψ V (ψ, ζ)V (χ, w) (w + 1)hχ −1 ζ w w 0 dζ dw = (ζ + 1)hψ V (Vn (ψ)χ, w)(ζ − w)−n−hψ (w + 1)hχ −1 ζ w 0 w n n+hψ −1 = hχ ≥n≥0 l=0 0 ∈ N (F0 ) . (−1)l hψ l+1−n dw w(w + 1) w+1 w l+1 (w + 1)hχ −n V (Vn (ψ)χ, w) (215) Because of (214), every element in N (F0 ) can be written as VR (φ) for a suitable φ, and (215) thus implies that [VL (ψ), N (χ)] ∈ N (F0 ); hence VL (ψ) defines an endomorphism of A(F0).
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. Since L0 is diagonalisable in every irreducible representation, it follows that the fusion product is necessarily not completely decomposable. This conclusion holds actually more generally whenever any correlation function contains a logarithm.
For c = cp,q the meromorphic Verma module has a null-vector at level (p − 1)(q − 1) (since it corresponds to r = s = 1), and (p−1)(q−1)/2 since the coefficient of L−2 Ω in the null-vector does not vanish [120–122], this (p−1)(q−1)/2 allows us to express L−2 Ω in terms of states in O(1)(F0), and thus shows that the dimension of A(1)(F0 ) is indeed (p − 1)(q − 1)/2. The minimal models include the unitary discrete series (138) for which we choose p = m and q = m + 1, but they also include non-unitary finite theories.