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

L.B. Drissi et al. / Nuclear Physics B 801 [FS] (2008) 316–345 329and alsoZ 2d =〈0|Γ + (1)Ψ − (q)|0〉,Z 3d =〈0|Ψ + (1)Ψ − (q)|0〉,(4.11)Z 0d =〈0|I id (1)Γ − (q)|0〉,Z 1d =〈0|Γ + (1)Γ − (q)|0〉,(4.12)where I id stands for the identity operator (of level zero). To get the Ω ± (z) operators, we require:(i) The Ω + (z) and Ω − (z) are local CFT 2 vertex operators that should obeyΩ − (x)Ω − (y) = Ω − (y)Ω − (y),Ω − (x)Ψ − (y) = Ψ − (y)Ω − (y),Ω − (x)Γ − (y) = Γ − (y)Ω − (y),Ω − (0) = 1,and similar relations for Ω + (x).(ii) We should also haveso thatΩ − (q) = q L 0Ω − (1)q −L 0Z 4d =〈0|Ψ + (1)q L 0Ω − (1)|0〉,Z 5d =〈0|Ω + (1)q L 0Ω − (1)|0〉,in analogy with the transfer matrix method used previously.(iii) We impose the commutation relations(4.13)(4.14)(4.15)Ψ + (1)Ω − (q) = G 4 (q)Ω − (q)Ψ + (1),Ω + (1)Ω − (q) = G 5 (q)Ω − (q)Ω + (1),(4.16)where G 4 (q) and G 5 (q) stand for the 4d- and 5d-generalized MacMahon functions given by,G 4 (q) =G 5 (q) =∞∏[( ) (k+1)! ] 1 (k−1)!2!1 − q k ,k=1∞∏[( ) (k+2)! ] 1 (k−1)!3!1 − q k .k=1A solution of these constraint relations is given by( −1( ∏ ∏−1(Ω − (1) =Γ− (1)q L )) )0q L 0,Ω + (1) =t 2 =−∞( ∞∏t 2 =0q L 0t 1 =−∞(∏ ∞(qL 0Γ + (1) ))) ,t 1 =0(4.17)(4.18)