An Introduction to Conformal Field Theory by M. Gaberdiel

By M. Gaberdiel

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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.

An arrow A −→ B indicates that the vertex B is in the image of A under the action of the Virasoro algebra. The representation R1,1 is not irreducible since the states that are obtained by the action of the Virasoro algebra from Ω form the subrepresentation H0 of R1,1 (that is actually isomorphic to the vacuum representation). On the other hand, R1,1 is not completely reducible since we cannot find a complementary subspace to H0 that is a representation by itself; R1,1 is therefore called an indecomposable (but reducible) representation.

The chiral n-point functions are largely determined in terms of the three-point functions of the theory. § Let us consider, as an example, the case of a 4-point function, where the four non-meromorphic fields φi ∈ Hmi , i = 1, . . , 4 are inserted at u1, . . , u4. ) In the limit in which u2 → u1 (with u3 and u4 far away), the 4-point function can be thought of as a three-point function whose non-meromorphic field at u1 ≈ u2 is the fusion product of φ1 and φ2 ; we can therefore write every 4-point function involving φ1 , .

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