By M. Gaberdiel
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Additional info for An Introduction to Conformal Field Theory
The analysis of  was adapted for the Wess-Zumino-Novikov-Witten models in . For a general affine algebra gˆ, the fusion rules can be determined from the so-called depth rule; in the specific case of g = su(2) at level k, this leads to j1 ⊗ j 2 = min(j1 +j2 ,k−j1 −j2 ) j, (232) j=|j1 −j2 | where j is integer if j1 + j2 is integer, and half-integer otherwise, and the highest weight representations are labelled by j = 0, 1/2, . . , k/2. A closed expression for the fusion rules in the general case is provided by the Kac-Walton formula [64, 132–134].
This theory is again bosonic provided the sign in (180) corresponds to the parity of dim Λ divided by 8. With this choice of (180) the orbifold theory defines another local meromorphic conformal field theory [71, 72, 111]. The most important example of this type is the orbifold theory associated to the Leech lattice for which the orbifold theory does not have any states of conformal weight one. This is the famous Monster conformal field theory whose automorphism group is the Monster group [12–14, 112], the largest simple sporadic group.
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. Furthermore, the operator product expansion of an h-current J a(z) and a vertex operator V (ψ, ζ) associated to ψ ∈ Hh is regular, and therefore the corresponding modes commute. This implies that the operator product expansion of two states in Hh only contain states that lie in Hh , and thus that we can define a meromorphic field theory whose space of states is Hh .
An Introduction to Conformal Field Theory by M. Gaberdiel