By Gaberdiel M.R.
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Extra resources for An introduction to conformal field theory (hep-th 9910156)
This idea has been made precise by Zhu  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 . 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∨ .