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

L.B. Drissi et al. / Nuclear Physics B 801 [FS] (2008) 316–345 333Indeed, we start from the definition of Z (p+1)d ,Z (p+1)d =〈0|Γ + (1)Γ (p+1)− (q)|0〉and express it, by using (5.5), as( ∏−1Z (p+1)d =〈0|Γ + (1)q L 0or equivalently likeZ (p+1)d =〈0|Γ + (1)( ∞∏l=1t=−∞Γ (p)− (1)qL 0Γ (p) (− ql )) |0〉.)|0〉(5.12)(5.13)(5.14)Then we commute Γ (p)− (ql ) to the left of Γ + (1) in Eq. (5.10), we get after some computations,∞∏ ∞∏[( ) (k+p−3)!1] (k−1)!(p−2)!Z (p+1)d =(5.15)1 − q l+k .l=1 k=1Setting s = (l + k), we can rewrite this relation as followsZ (p+1)d =∞∏s=1 k=1s∏[( ) (k+p−3)! ] 1 (k−1)!(p−2)!1 − q s .(5.16)At first sight, this expression seems different from the desired result; however explicit computationleads exactly to the right result; thanks to the combinatorial identity,s∑ (k + p − 3)! (s + p − 2)!= , p 2,(5.17)(k − 1)!(p − 2)! (s − 1)!(p − 1)!k=1which is showed in Appendix B.These computations give an explicit proof for the derivation of the expression of generalizedMacMahon function. Thanks to the “transfer matrix method” and to the hierarchy of level pvertex operators Γ (p)± Eq. (5.5).6. G n (q) as (n + 1)-point correlation functionSo far we have seen that n-dimensional generalization of MacMahon function G n (q) withn 2, can be interpreted as 2-point correlation functions of some composite vertex operators.We have also seen that there are different, but equivalent ways, to express G n (q) as 2-pointcorrelation functions. Using Γ ± (r) (s)and Γ ± vertex operators, one can check that for any positivedefinite integers r and s such that r + s − 1 = n,wehave,G n (q) =〈0|Γ (n−s+1)+ (1)Γ − (s)(q)|0〉, 1 s n, n 1.The 2( r+s−22) + 1 possibilities are all of them equal to each other. This diversity in definingG n (q) suggests us to look for a more refined definition of it. We have found that the adequateway to define G n (q) is like a (n + 1)-point correlation function as given below,(6.1)G n (q) = G n+1 (x 0 ,x 1 ,x 2 ,...,x n )(6.2)