Contributions à l'Etude du Vertex Topologique en Théorie ... - Toubkal

L.B. Drissi et al. / Nuclear Physics B 801 [FS] (2008) 316–345 339Moreover, using the fact that 3d partitions Π (3) may themselves be sliced in terms of 2dpartitions, one can usually bring the correlation function C ΛΣΥ Ψ to the form,〈〈ς,τ,υ‖A ′ +[(λ,μ,ν); (ζ,η,θ)]A′−[(λ t ,μ t ,ν t) ; ( ζ t ,η t ,θ t)] ‖α, β, γ 〉〉,(7.13)where ‖ϑ, σ, ϱ〉〉 is a 3d partition boundary state expressed in terms of 2d partitions |ϑ〉⊗|σ 〉⊗|ϱ〉. In the particular case α = β = γ =∅and ς = τ = υ =∅, the correlation function becomes〈〈∅, ∅, ∅‖A ′ +[(λ,μ,ν); (ζ,η,θ)]A′−[(λ t ,μ t ,ν t) ; ( ζ t ,η t ,θ t)] ‖∅, ∅, ∅〉〉.In the special case ζ = η = θ = λ = μ = ν =∅, the above quantity simplifies as(7.14)〈〈∅, ∅, ∅‖A ′ + [∅]A′ − [∅]‖∅, ∅, ∅〉〉.(7.15)From this general relation, we see that the MacMahon function G 4 (q) =〈0|Ψ + (1)Ω − (q)|0〉Eq. (4.10) appears as a very particular correlation function and then cannot be the generatingfunctional of all possible 4d partitions.AcknowledgementsThis research work is supported by the program Protars III D12/25. H.J. would like to thankICTP for kind hospitality where part of this work has been done. The authors thank B. Szendroifor helpful suggestion.Appendix A. Vertex operators Γ (n)± (x)In this appendix we describe the vertex operators Γ ± (n) (x) and their commutation relationsalgebra.We first study the level n vertex operators Γ ± (n) (x) and their main properties starting byΓ ± (2) = Ψ ±. Then, we give their algebra.A.1. Level 2 vertex operatorTo begin notice that the operators Ψ ± (1) Eq. (3.11), denoted also as Γ ± (2) (1), can be put in theform,[( s∏(Ψ − (1) = lim Γ − qt )) ]q sL 0,s→∞t=0[ ( s∏Ψ + (1) = lim q sL (0Γ + q−t ))] .(A.1)s→∞t=0Using the expression of Γ ± (z) Eq. (2.11), we can rewrite Ψ ± (1) as follows:( −1)∏∞∏Ψ − (y) = Γ − (y)q L (0= Γ − q k y ) ,t=−∞k=0( ∞)∏∞∏Ψ + (x) = q L (0Γ + (x) = Γ + q −k x ) ,t=0t=0(A.2)