An introduction to conformal field theory (hep-th 9910156) by Gaberdiel M.R. PDF

By Gaberdiel M.R.

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Let us introduce, following (209) and (213), the notation VL (ψ) ≡ V (0)(ψ) VR (ψ) ≡ Vc(0)(ψ) N (ψ) = V (1)(ψ) = Vc(1)(ψ) . We also denote by N (F0) the vector space of operators that are spanned by N (ψ) for ψ ∈ F0 ; then O(F0 ) = N (F0 )F0 . Finally it follows from (82) that VL (ψ)Ω = VR (ψ)Ω = ψ. The equations (209) and (213) suggest that the modes VL (ψ) and VR (χ) commute up to an operator in N (F0 ). In order to prove this it is sufficient to consider the case where ψ and χ are both eigenvectors of L0 with eigenvalues hψ and hχ , respectively.

If we choose B = 0, then because of (244), F has a logarithmic branch cut at x = 1. In terms of the representation theory this logarithmic behaviour is related to the property of the fusion product of µ with itself not to be completely reducible: by considering a suitable limit of z1 , z2 → ∞ in the above 4-point function we can obtain a state Ω satisfying 1 Ω (∞) µ(z)µ(0) = z 4 A + B log(z) , (245) Conformal Field Theory 51 where A and B are constants (that depend now on Ω ). We can therefore write 1 µ(z)µ(0) ∼ z 4 ω(0) + log(z)Ω(0) , (246) where Ω (∞)ω(0) = A and Ω (∞)Ω(0) = B.

178) These act on an irreducible representation space U of the algebra γ i γ j = (−1)ei ·ej γ j γ i , (179) where ei , i = 1, . . , n is a basis of Λ, and where for χ ∈ U , cir χ = 0 if r > 0. The actual orbifold theory consists then of the states in the untwisted HΛ and the twisted sector HΛ that are left invariant by θ, where the action of θ on HΛ is given as in (177), and on HΛ we have θcir θ = −cir θ|U = ±1 . (180) The generators of the Virasoro algebra act in the twisted sector as Lm = 1 2 n : cir cim−r : + i=1 r∈Z+ 1 2 n δm,0 .

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An introduction to conformal field theory (hep-th 9910156) by Gaberdiel M.R.


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