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

332 L.B. Drissi et al. / Nuclear Physics B 801 [FS] (2008) 316–345or equivalently by using Eq. (3.11), like( −1[( ∏ ∏−1Z 3d =〈0|Γ + (1)q L 0t 2 =−∞t 1 =−∞(Γ− (q)q L 0 )) q L 0This relation as well as Eqs. (4.4)–(4.6) suggest us the structure of the p-dimensional partitionfunction Z pd in terms of CFT 2 ’s vertex operators Γ ± . For doing so, we need to introduce thefollowing hierarchy of local vertex operators( −1[( ∏Γ − (n+1)∏−1∏−1((1) = ···Γ− (1)q L )) ] )0q L 0···q L 0(5.3)for n 1, together withΓ (0)t n =−∞t 2 =−∞− = I id, Γ −(1) (z) = Γ −(z).Eq. (5.3) can be also defined as follows,Γ (n+1)− (1) =−1∏t=−∞t 1 =−∞( (n) ) Γ − (1)qL 0, n 1.A similar relation can be written down for Γ + (n+1) (1). TheΓ (p)− , referred to as the level p vertexoperator, obey quite similar relations that the ones associated to Γ − (1), in particularΓ (p)− (q) = qL 0Γ (p)− (1)q−L 0, p 0.])|0〉.(5.2)(5.4)(5.5)More details, concerning these high level operators, are presented in Appendix A.Based on the preceding results realized for lower dimensions, it follows that the p-dimensional partition functions Z pd can be defined as,Z pd =〈0|Γ + (1)Γ (p)− (q)|0〉, p 0.This relation, which has been explicitly checked for p = 0, 1, 2, 3, 4 and 5, reads also asZ pd =〈0|Γ + (1)q L 0Γ (p)− (1)|0〉.Commuting Γ (p)− (q) to the left of Γ +(1), we can show by induction that for p 2Z pd =∞∏[( ) (k+p−3)! ] 1 (k−1)!(p−2)!1 − q k .k=1Proof by inductionWe suppose that Eq. (5.9) holds for level p; then prove that it holds as well for level (p + 1);that is,Z (p+1)d =〈0|Γ + (1)Γ (p+1)− (q)|0〉,and find that it is given by∞∏[( ) (k+p−2)! ] 1 (k−1)!(p−1)!Z (p+1)d =1 − q k .k=1(5.6)(5.7)(5.8)(5.9)(5.10)(5.11)