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

326 L.B. Drissi et al. / Nuclear Physics B 801 [FS] (2008) 316–345(i) Γ ± (z) and Ψ ± (z) are related by the mappingz ±n →z±n1 − q n .(3.24)Since for q → 0, Γ ± (z) and Ψ ± (z) coincide, it follows that the operators Ψ ± (z) can be interpretedas a q-deformation Γ ± (z).(ii) Γ ± (z) and Ψ ± (z) share most of the basic quantum properties since both of them involvethe same Kac–Moody mode operators J ±n ,(3) TranslationsThe operator q L 0 acts also as a translation operator on Ψ ± (z) in the same manner like forΓ ± (z)q L 0Ψ ± (z)q −L 0= Ψ ± (qz).(3.25)This property allows us to define 4d-generalization from 3d one in quite similar manner as wehave done in going from 2d to 3d. We will come back to this feature later.(4) Z 3d as a “2-point correlation” functionUsing Ψ ± (1) vertex operators, the partition function Z 3d can be put in the simplest formZ 3d =〈0|Ψ + (1)q L 0Ψ − (1)|0〉.By help of the identity q L 0Ψ − (1)q −L 0 = Ψ − (q) Eq. (3.25),wealsohave(3.26)Z 3d =〈0|Ψ + (1)Ψ − (q)|0〉,(3.27)where Z 3d appears as just the 2-point correlation function of the level 2 vertex operators Ψ + (1)and Ψ − (q). It happens that Eq. (3.27) is not the unique way to define Z 3d . Let us comment brieflyaspects of this issue; general results will be given in Sections 5 and 6.(i) Eq. (3.27) can be also expressed as follows( −1)∏Z 3d =〈0|Γ + (1)q L 0Ψ − (1)q L 0|0〉.(3.28)t=−∞This expression will allow us to get the definition of higher dimensional generalizations ofMacMahon function; see Eq. (5.2). This relation can be put in the simple formZ 3d =〈0|Γ + (1)q L 0Ω − (1)|0〉,or equivalentlyZ 3d =〈0|Γ + (1)Ω − (q)|0〉,where we have set( s∏(Ω − (1) = lim Ψ − qk )) q sL 0.s→∞t=0(3.29)(3.30)(3.31)This local vertex operator should be thought of as the level 3 of the hierarchy we have refereedto earlier; i.e.Ω ± (1) = Γ (3)± .(3.32)