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By Gaberdiel M.R.

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One can analyse the fusion product of µ with itself using the comultiplication formula, and this allows one to determine the structure of the resulting representation R1,1 in detail [143]: the representation is generated from a highest weight state ω satisfying L0 ω = Ω , L0 Ω = 0 , Ln ω = 0 for n > 0 (252) by the action of the Virasoro algebra. The state L−1 Ω is a null-state of R1,1 , but L−1 ω is not null since L1 L−1 ω = [L1 , L−1 ]ω = 2L0 ω = 2Ω. Schematically the representation can therefore be described as × R1,1 × ■ ❅ ✠ ❅ ■ ❅ h=1 • ■ ❅ ❅ ❅ ✠ ❅ •✛ • Ω ω h=0 Conformal Field Theory 52 Here each vertex • denotes a state of the representation space, and the vertices × correspond to null-vectors.

197) Conformal Field Theory 40 Indeed, if |j, j is a Virasoro highest weight state with J0+ |j, j = 0 and H0 |j, j = j|j, j , then + = |j, j , J1− J−1 |j, j + + J−1 |j, j , J−1 |j, j = (k − 2j) |j, j , |j, j , (198) and if the representation is unitary, this requires that (k − 2j) ≥ 0, and thus that j = 0, 1/2, . . , 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.

225) Here the action of the modes of the meromorphic fields on φi or φj is defined as in (187). The comultiplication depends on u1 and u2, and for the modes of a field ψ of conformal weight h, it is explicitly given as n ∆u1 ,u2 (Vn (ψ)) = m=1−h n+h−1 m+h−1 n + ε1 l=1−h un−m (Vm (ψ) ⊗ 1l) 1 n+h−1 l+h−1 un−l (1l ⊗ Vl (ψ)) 2 (226) † An alternative (more mathematical) definition of this tensor product was developed by Huang & Lepowsky [127–130]. e. V (χj , ζj )Vn (ψ) φi (u1)φj (u2 ) j V (χj , ζj ) ∆(1)(Vn (ψ))φi (u1) ∆(2)(Vn (ψ))φj (u2) = , (228) j where each χj can be a meromorphic or a non-meromorphic field and we have used the notation of (225).

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