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By Gaberdiel M.R.

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If we choose B = 0, then because of (244), F has a logarithmic branch cut at x = 1. In terms of the representation theory this logarithmic behaviour is related to the property of the fusion product of µ with itself not to be completely reducible: by considering a suitable limit of z1 , z2 → ∞ in the above 4-point function we can obtain a state Ω satisfying 1 Ω (∞) µ(z)µ(0) = z 4 A + B log(z) , (245) Conformal Field Theory 51 where A and B are constants (that depend now on Ω ). We can therefore write 1 µ(z)µ(0) ∼ z 4 ω(0) + log(z)Ω(0) , (246) where Ω (∞)ω(0) = A and Ω (∞)Ω(0) = B.

The analysis of [4] was adapted for the Wess-Zumino-Novikov-Witten models in [60]. For a general affine algebra gˆ, the fusion rules can be determined from the so-called depth rule; in the specific case of g = su(2) at level k, this leads to j1 ⊗ j 2 = min(j1 +j2 ,k−j1 −j2 ) j, (232) j=|j1 −j2 | where j is integer if j1 + j2 is integer, and half-integer otherwise, and the highest weight representations are labelled by j = 0, 1/2, . . , k/2. A closed expression for the fusion rules in the general case is provided by the Kac-Walton formula [64, 132–134].

The actual (local) amplitudes (that are linear combinations of products of the chiral and anti-chiral amplitudes) can then be determined from these by the conditions that (i) they have to be local, and (ii) the operator product expansion (7) is indeed associative. These constraints define the so-called bootstrap equations [4]; in practice they are rather Conformal Field Theory 55 difficult to solve, and explicit solutions are only known for a relatively small number of examples [41, 156–164]. The chiral n-point functions are largely determined in terms of the three-point functions of the theory.

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

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