By Gaberdiel M.R.
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Extra info for An introduction to conformal field theory (hep-th 9910156)
E. 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.
4 as k Nm 1 m2 αk,i Φk,i 12 (u1 , u2 ) φ3 (u3 )φ4 (u4 ) , φ1(u1 )φ2(u2 )φ3(u3 )φ4(u4 ) = k (266) i=1 Φk,i 12 (u1 , u2 ) ∈ k which Nm 1 m2 where Hk , αk,i are arbitrary constants, and the sum extends over those k for ≥ 1. The number of different three-point functions involving k k∨ , and the number of different solutions Φ12 (u1, u2) ∈ Hk , φ3 and φ4, is given by Nm 3 m4 is therefore altogether ∨ k k Nm Nm . e. the solutions in terms of which we have expanded (266)) are characterised by the condition that they can be approximated by a product of three-point functions as u2 → u1.
2. An Illustrative Example It should be stressed at this stage that the condition to be a representation of the meromorphic conformal field theory is usually stronger than that of being a representation of the Lie algebra (or W-algebra) of modes of the generating fields. 2, the latter condition Conformal Field Theory 39 means that the representation space has to be a representation of the affine algebra (130), and for any value of k, there exist infinitely many (non-integral) representations. On the other hand, if k is a positive integer, the meromorphic theory possesses null-vectors, and only those representations of the affine algebra are representations of the meromorphic conformal field theory for which the null-fields act trivially on the representation space; this selects a finite number of representations.