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

318 L.B. Drissi et al. / Nuclear Physics B 801 [FS] (2008) 316–345(1) Give a conformal field theoretical derivation of the conjectured d-dimensional generalizedMacMahon function G d expressed by the following formula,G d (q) =∞∏ [(1 − qk ) − (k+d−3)! ] (k−1)!(d−2)!, d 2,k=1together with the two special ones filling the hierarchy,G 1 (q) = 11 − q , G 0(q) = 1.Recall that in combinatorial analysis, the function G 3 (q) can be defined as the generating functionalof 3-dimensional partitions Π (3) extending the usual 2d-partitions μ = Π (2) to higher3-dimensions. Refined studies regarding G 4 function have revealed that it is not the generatingfunctional of 4d partitions [18,28,33].To fix the ideas, it is interesting to recall that expanding G 2 (q) as a q n power series like,∞∏( ) 1∞∑G 2 (q) =1 − q k = p 2 (n)q n ,k=1n=0one gets the number p 2 (n) of 2d-partitions (Young diagrams) containing n boxes. From thisview, G 2 can be physically interpreted as the exact partition function Z 2 = Tr(q H ) of a twodimensionalstatistical physics system with,(q = exp− 1KT),and energy spectrum E k = k. HereT is the absolute temperature and the constant K is theBoltzmann one. For instance, G 2 (q) is the partition function of the c = 1 free Bose gas. There,the Hamiltonian is given by H = ¯hω ∑ kN k with N k = a + k a k being the operator number ofparticles and energy spectrum E k = ¯hωk.Similar expansions can be also made for G d (q) which then read as followsG d (q) =∞∑p d (n)q n , d 3.n=0For the case d = 3, the number p 3 (n) is precisely the number of 3d-partitions; but for d = 4, thenumber p 4 (n) is not the total number of 4d-partitions as it has been explicitly checked in [28].(2) The second objective of the present study is to show that G d (q) can be remarkably interpretedas a (d +1)-point correlation function G d+1 of some q-deformed vertex operators O j (x j ),i.e.(1.1)G d (q) = G d+1 (x 0 ,x 1 ,x 2 ,...,x d )with x j = q j , j = 0,...,d; and(1.2)G d+1 =〈0|O 0 (x 0 )O 1 (x 1 )O 2 (x 2 ) ···O d (x d )|0〉.The O j (x j )’s will be determined in terms of the usual vertex operators Γ ± = exp(Φ ± ) of thec = 1 two-dimensional bosonic conformal field theory [34]; but also others, denoted like Γ (p)± ,involving q-deformed QFT 2 . This result gives:(1.3)