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

L.B. Drissi et al. / Nuclear Physics B 801 [FS] (2008) 316–345 341(ii) Computation using directly Eqs. (A.3). We get,( ∑ 1 y n x −n )Ψ + (x)Ψ − (y) = expn (1 − q n ) 2 Ψ − (y)Ψ + (x).n1By comparing the two expressions (A.6) and (A.11), we get the following identity( ∑ 1 y n x −n ) ∞∏(expn (1 − q n ) 2 = 1 − q s y ) −(s+1),xn1s=0or equivalently like,∑ 1 y n x −n ∞n (1 − q n ) 2 =− ∑[ ((s + 1) ln 1 − q s y )].xn1s=0(A.11)(A.12)A.2. Generic q-deformed operatorsHere we give the expressions of the generic q-deformed operators and some useful propertiesof their algebra.The starting point is the vertex operators(Γ − (z) = exp i ∑ )(1n zn J −n , Γ + (z) = expn>0−i ∑ n>0)1n z−n J n(A.13)and the aim is:(1) compute for n 1, the following hierarchy of composite vertex operators( ∞[Γ − (n+1) ∏ ∞(∏ ∞) ] )∏(z) = ··· Γ − (z)q L 0q L 0···q L 0.t n =1t 2 =1t 1 =1(A.14)For n = 0, we have just Γ −(1) (z) = Γ −(z). Similar quantities can be written down for Γ + (n+1) (z);we shall not report them here.(2) Derive the identity (6.30).For these purposes, we proceed by using inductive method:q-deformed vertex operators Γ (2)−(a) Case Γ −(2) (z)In this case, we have(3)(z) and Γ − (z):Γ (2)−Notice thatΓ (2)−∞ (z) = ∏( Γ− (z)q L ) 0.t=1(z) = lims→∞t=1s∏ (Γ− (z)q L ) 0.(A.15)(A.16)