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

(∏ ∞(Υ − (q) = Ω − qk ))k=1which should be thought of as Υ − = Γ (4)− .L.B. Drissi et al. / Nuclear Physics B 801 [FS] (2008) 316–345 3315d caseSimilarly, we have for the 5d-generalization (4.15),(∏ ∞)Z 5d =〈0| q L 0Ψ + (1) q L 0Ω − (1)|0〉,t=0(4.26)(4.27)where we have substituted Ω + in terms of product of Ψ + (4.19). Next using the fact that q L 0 actsas a translation operator, we can put Z 5d as follows(∏ ∞(Z 5d =〈0| Ψ + q−l )) Ω − (1)|0〉.(4.28)l=1Then using the identity( ) [1 ∏ ∞ ( ) s(s+1) ] ( )1 21Ψ + Ω − (1) =x1 − xq s Ω − (1)Ψ + ,xwe obtainZ 5d =∞∏ ∞∏[(s=1 l=1s=1) s(s+1)121 − q l+sThe next step is to put it in the form∞∏ k∏[( ) s(s+1)1 2Z 5d =1 − q kwhich givesZ 5d =k=1 s=1k=1],].∞∏[( ) k(k+1)(k+2) ]161 − q k .In getting this relation, we have used the identityk∑ s(s + 1) k(k + 1)(k + 2)= ,26s=1(4.29)(4.30)(4.31)(4.32)(4.33)proved in Appendix B. Here also we have different, but equivalent, ways to define Z 5d . Later on,we will give the exact numbers of ways for generic Z pd .5. Result by inductionFirst notice that the expression (3.27) of the partition function Z 3d can be also put in the formZ 3d =〈0|Γ + (1)q L 0( −1 ∏t 2 =−∞(Ψ− (1)q L 0 )) |0〉(5.1)