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

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By Gaberdiel M.R.

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E. one that only depends on the z¯i. However, for n = 0, 1, 2, 3, the sum contains only one term since the functional form of the relevant chiral (and anti-chiral) amplitudes is uniquely determined by M¨obius symmetry. The zero-point functions are simply products of meromorphic amplitudes, and the one-point functions vanish. The two-point functions are usually non-trivial, and they define, in essence, the different representations of the meromorphic and the antimeromorphic subtheory that are present in the theory.

If the representation is untwisted, we can expand the meromorphic fields in terms of their modes as in (80). In this way we can then define the action of Vn (ψ) on the non-meromorphic state |φ , and thus on arbitrary states of the form Vn1 (ψ1) · · · VnN (ψN )|φ . 4, the commutation relations of these modes (90) can be derived from the operator product expansion of the corresponding fields. Since the representation amplitudes (183) preserve these in the sense of (186), it follows that the action of the modes on the states of the form (187) also respects (90), at least up to null-states that vanish in all amplitudes.

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.

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