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

L.B. Drissi et al. / Nuclear Physics B 801 [FS] (2008) 316–345 327(ii) Along with the two representations (3.27) and (3.30) the partition function Z 3d can beexpressed as well like( ) 1Z 3d =〈0|Ω + Γ − (1)|0〉,(3.33)qwhere we have used the correlation of Ω + (x) and Γ − (y) rather than Γ + (x) and Ω − (y).(iii) The diversity in expressing Z 3d as a 2-point correlation function, let us suspect that Z 3dcould be expressed as a more basic objects. In exploring this idea, we have found that the adequateinterpretation of Z 3d (q) is as a special 4-point correlation functionZ 3d (q) = G 4 (x 0 ,x 1 ,x 2 ,x 3 ),(3.34)of vertex operators O j (x j ) involving different Γ (p)± levels,G 4 =〈0|O 0 (x 0 )O 1 (x 1 )O 2 (x 2 )O 3 (x 3 )|0〉.(3.35)To fix the ideas keep in mind the two following:(α) the vertex operator O 0 (x 0 ) stands for Γ + (1) and the other operators will be explicitlygiven in Section 6;seeEqs.(6.34).(β) the observed diversity in defining Z 3d (q) corresponds just to decomposing (3.35) by usingWick theorem combined with Eqs. (2.18)–(2.19).4. Extension to 4d and 5dWe first show that the leading terms of the generalized MacMahon function can be realized as2-point functions of some vertex operators of c = 1 2d conformal field theory.Then, we use this feature to derive the general formula for G d (q). In Section 5, we considerthe interpretation of G d (q) as (d + 1)-points correlation function G 4 (x 0 ,x 1 ,...,x d ) involvingvertex operators O j (x j ) as in Eq. (1.2).4.1. Z 1d and Z 2d as 2-point functionsBefore studying 4d and 5d generalizations, it is interesting to start by revisiting the 1d and2d cases. This is an important thing for getting the full picture on the conjectured MacMahonfunction G d .We start by noting the two following:(1) Recall that the 1d-MacMahon function corresponds to,Z 1d = 1(4.1)1 − q .This function can be exactly interpreted as the two-point correlation 4Z 1d = G 2 = G 2 (z 0 ,z 1 ),(4.2)of the vertex operators Γ + (1) and Γ − (q) as shown below,Z 1d =〈0|Γ + (1)Γ − (q)|0〉.(4.3)4 Z pd is the MacMahon function G pd ; it should not be confused with its interpretation as (p + 1)-points correlationfunction G p+1 = G(x 0 ,x 1 ,...,x p ) to be studied in Section 6; see also Eqs. (1.1).