By Gaberdiel M.R.

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176) together with additional twisted sectors, one for each conjugacy class in G. Generically the meromorphic subtheory of the resulting theory consists of those meromorphic fields of the original theory that are invariant under G, but in general it may also happen that some of the twisted sectors contain additional meromorphic fields. The construction of the twisted sectors is somewhat formal in general, and so is the proof that the resulting local theory is always modular invariant. 4. Let us for simplicity consider the case where G = Z2 = {1, θ}, and the action on HΛ is determined by θH i (z)θ = −H i (z) , θV (x, z)θ = V (−x, z) , θΩ = Ω .

Each such representation is cp,q = 1 − †† Incidentally, A(1)(F0 ) also has the structure of an abelian algebra; the significance of this algebra is however not clear at present. Conformal Field Theory 45 the irreducible quotient space of a Verma module generated from a highest weight state with conformal weight h, and the allowed values for h are h(r,s) = (rp − qs)2 − (p − q)2 , 4pq (220) where 1 ≤ r ≤ q − 1 and 1 ≤ s ≤ p − 1, and (r, s) defines the same value as (q − r, p − s). Each of the corresponding Verma modules has two null-vectors at conformal weights h + rs and h + (p − r)(q − s), respectively, and the actual Fock space is the quotient space of the Verma module by the subspace generated by these two null-vectors [68, 69, 121].

Their functional form is uniquely determined, and one of the essential pieces of information is therefore whether the corresponding amplitudes can be non-trivial or not; this is encoded in the so-called fusion rules. The definition of the fusion rule is actually slightly more complicated since there can also be non-trivial multiplicities. In fact, the problem is rather analogous to that of decomposing a tensor product representation (of a compact group, say) into irreducibles. Because of the M¨obius covariance of the amplitudes, it is sufficient to consider the amplitudes of the form φk (∞) V (ψ1, z1) · · · V (ψl , zl ) φi (u1)φj (u2 ) , (223) where the three non-meromorphic fields are φi , φj and φk , and we could set u1 = 1 and u2 = 0, for example.

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

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