By M. Gaberdiel
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Extra info for An Introduction to Conformal Field Theory
Let us denote by O(F0 ) the subspace of F0 that is generated by states of the form V (N )(ψ)χ with N > 0, and define the quotient space A(F0) = F0 /O(F0 ). The above then implies that we can associate a zero mode (acting on a highest weight state) to each state in A(F0). We can write (202) in terms of modes as h V (N ) (ψ) = n=0 h V−n−N (ψ) , n (204) where ψ has conformal weight h, and it therefore follows that V (1)(ψ)Ω = V−h−1 (ψ)Ω + hV−h (ψ)Ω = (L−1 + L0 )ψ . (205) Thus O(F0 ) contains the states in (199).
The following explanation of Zhu’s work follows closely . In a first step we determine the subspace of states whose zero modes always vanish on Virasoro highest weight states. This subspace certainly contains the states of the form (L−1 + L0)ψ, where ψ ∈ F0 is arbitrary, since (108) implies that V0 ((L−1 + L0 )ψ) = V0 (L−1 ψ) + hV0 (ψ) = 0 . (199) Furthermore, the subspace must also contain every state whose zero mode is of the form V0 (χ)V0 ((L−1 + L0 )ψ) or V0 ((L−1 + L0)ψ)V0 (χ). In order to describe states of this form more explicitly, it is useful to observe that if both φ and φ¯ are Virasoro highest weight states ¯ (ψ, 1)|φ = φ|V ¯ l (ψ)|φ φ|V l ¯ 0 (ψ)|φ , = φ|V (200) since the highest weight property implies that Vl (ψ)|φ = 0 for l > 0 and similarly, using ¯ l (ψ) = 0 for l < 0.
A priori it is not clear whether the actual product space may not be even smaller. However, the fusion rules for a number of models have been calculated with this ‡ It is a priori ambiguous whether a given vector in a representation space is fermionic or not. However, once a convention has been chosen for one element, the fermion number of any element that can be obtained from it by the action of the modes of the meromorphic fields is well defined. Conformal Field Theory 48 definition [126, 131], and the results coincide with those obtained by other methods.
An Introduction to Conformal Field Theory by M. Gaberdiel