By M. Gaberdiel

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**Additional resources for An Introduction to Conformal Field Theory**

**Example text**

E. a linear combination of states of the form (84) that vanishes in every meromorphic amplitude) then any amplitude (183) involving N also vanishes; this is Conformal Field Theory 37 implicit in the above since the operator product expansion of the meromorphic conformal field theory is only determined up to such null-fields by the meromorphic amplitudes. We call a representation untwisted if the amplitudes (183) are single-valued as zi encircles the origin or infinity; if this is not the case for at least some of the meromorphic fields the representation is called twisted.

A closed expression for the fusion rules in the general case is provided by the Kac-Walton formula [64, 132–134]. Similarly, the fusion rules have been determined for the W3 algebra in [135], the N = 1 superconformal minimal models in [136] and the N = 2 superconformal minimal models in [137, 138]. For finite theories the fusion rules can also be obtained by performing the analogue of Zhu’s construction in each representation space; this was first done (in a slightly different language) by Feigin & Fuchs for the minimal models [139], and later by Frenkel & Zhu for general vertex operator algebras [61].

We can generalise this construction directly to the case where instead of the affine theory associated to gˆ, we consider an arbitrary conformal field theory H (with stressˆ Then energy tensor L) that contains, as a subtheory, the affine theory associated to h. by the same arguments as above K = L − Lh (173) Conformal Field Theory 33 ˆ (and thus with Lh ), and therefore satisfies a Virasoro algebra with commutes with h central charge cK = c−ch . By construction, the Virasoro algebra Km leaves the subspace of states ˆ with n ≥ 0 Hh = ψ ∈ H : Jnaψ = 0 for every Jna ∈ h (174) invariant.

### An Introduction to Conformal Field Theory by M. Gaberdiel

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