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By Gaberdiel M.R.

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Extra resources for An introduction to conformal field theory (hep-th 9910156)

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This idea has been made precise by Zhu [16] who constructed an algebra, now commonly referred to as Zhu’s algebra, that describes the algebra of zero modes modulo zero modes of null-vectors, and whose representations are in one-to-one correspondence with those of the meromorphic conformal field theory. The following explanation of Zhu’s work follows closely [48]. 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 .

Let us denote by Lgm and Lhm the modes ˆ respectively. If J a is a of the Sugawara fields (134) of the affine algebras gˆ and h, n ˆ ⊂ gˆ, then generator of h a [Lgm , Jna] = −nJm+n and a [Lhm , Jna ] = −nJm+n (168) since Jna are the modes of a primary field of conformal weight h = 1. ) It then follows that Km = Lgm − Lhm (169) Lgm = Lhm + Km , (170) ˆ and therefore with the modes Lh (which are commutes with every generator Jna of h, m bilinear in Jna). We can thus write Lgm as the sum of two commuting terms and since both Lgm and Lhm satisfy the commutation relations of a Virasoro algebra, it follows that this is also the case for Km , where the corresponding central charge is cK = c g − c h .

It is then natural to define ∨ Nijk ≡ Nijk , (259) which is manifestly symmetric under the exchange of i, j and k. The fusion product is also associative, m Nijk Nkl = k m k Nik Njl . (260) k If we define Ni to be the matrix with matrix elements (Ni )kj ≡ Nijk , (261) then (260) can be rewritten as (Ni )kj (Nl )m k = k (Nl )kj (Ni )m k , k (262) Conformal Field Theory 54 where we have used (259). Thus the matrices Ni commute with each other. e. they commute with their adjoint (or transpose) since N†i = Ni∨ .

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