By Gaberdiel M.R.
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Additional resources for An introduction to conformal field theory (hep-th 9910156)
N with r ∈ Z + 12 , satisfying the commutation relations [cir , cjs ] = rδ ij δr,−s . (178) These act on an irreducible representation space U of the algebra γ i γ j = (−1)ei ·ej γ j γ i , (179) where ei , i = 1, . . , n is a basis of Λ, and where for χ ∈ U , cir χ = 0 if r > 0. The actual orbifold theory consists then of the states in the untwisted HΛ and the twisted sector HΛ that are left invariant by θ, where the action of θ on HΛ is given as in (177), and on HΛ we have θcir θ = −cir θ|U = ±1 .
K/2. As it turns out, this is also sufficient to guarantee unitarity. In general, however, the constraints that select the representations of the meromorphic conformal field theory from those of the Lie algebra of modes cannot be understood in terms of unitarity. 3. Zhu’s Algebra and the Classification of Representations The above analysis suggests that to each representation of the zero modes of the meromorphic fields for which the zero modes of the null-fields vanish, a highest weight representation of the meromorphic conformal field theory can be associated, and that all highest weight representations of a meromorphic conformal field theory can be obtained in this way .
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 , and later by Frenkel & Zhu for general vertex operator algebras . ) One of the advantages of the approach that we have adopted here is the fact that structural properties of fusion can be derived in this framework . For each representation Hj , let us define the subspace Fj− of the Fock space Fj to be the space that is spanned by the vectors of the form V−n (ψ)Φ where Φ ∈ Fj and n ≥ h(ψ).