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

320 L.B. Drissi et al. / Nuclear Physics B 801 [FS] (2008) 316–345(2) c = 1 conformal modelIn the 2D conformal field theoretic realization of Eqs. (2.1)–(2.3), one distinguishes two freefield theoretic realizations of the c = 1 conformal representation:(i) The free bosonic realization using a single real (chiral) boson Φ(z) with the usual two-pointcorrelation function,Φ(z 1 )Φ(z 2 ) =−ln(z 1 − z 2 ) + regular terms.(ii) The free fermionic realization using a complex one component fermion ψ(z). In this case,the two-point correlation function that have a singular term is,ψ ∗ (z 1 )ψ(z 2 ) = 1z 1 − z 2+ regular terms.The two-point functions ψ(z 1 )ψ(z 2 ) and ψ ∗ (z 1 )ψ ∗ (z 2 ) are regular.The U(1) Kac–Moody current J(z)is given, in the bosonic representation, by:J(z)= ∂Φ(z)i∂z ,while it has the following form J(z) =:iψ ∗ (z)ψ(z): in terms of fermions. Below, we shallmainly focus on the bosonic case; the link with fermionic representation can be done by usingbosonization ideas.Expanding the 2d chiral scalar field as(2.4)(2.5)(2.6)Φ(z) = ∑ n∈Zz −n Φ nand rearranging it as Φ(z) = Φ − (z) + Φ 0 + Φ + (z), we can write the above expansion as,Φ − (z) = i ∑ n>01n zn J −n , Φ + (z) =−i ∑ 1n z−n J nn>0(2.7)(2.8)wherewehaveusedΦ n = 1(2.9)in J n, n∈ Z ∗ .This identity follows directly by comparing Eqs. (2.6)–(2.7) and (2.2). Notice also that the zeromode Φ 0 acts trivially; it will be ignored in follows.(3) Vertex operators: Level 1There are various local field vertex operators that we will encounter in this present study. Thesimplest ones, named as level 1,aregivenbyΓ ± (z) = exp ( Φ ± (z) ) , z∈ C.(2.10)The other vertex operators Γ (p)± (z), to be introduced later on, will be named as level p vertexoperators. Substituting Φ ± (z) by their expression (2.8), thelevel 1 vertex operators (Γ ± (z) ≡Γ ± (1) (z)) read also as follows(Γ − (z) = exp i ∑ )1n zn J −n ,n>0