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

L.B. Drissi et al. / Nuclear Physics B 801 [FS] (2008) 316–345 337∏G p (q) =〈0|Γ + (1)By comparing with,we obtain(p−1 pkk=0∏l k =1Γ −(k+1) (qk+1 )) |0〉, p 1.G p+1 =〈0|O 0 (x 0 )O 1 (x 1 )O 2 (x 2 ) ···O p (x p )|0〉,(6.32)(6.33)O 0 (x 0 ) = Γ + (1), O 1 (x 1 ) = Γ − (1)( (q), ∏ pj[ (j)( O j (x j ) = Γ qj )]) , j 2.(6.34)l j =1−For more details on the derivation of this relation, see Appendix A: Eqs. (A.27)–(A.37).Eqs. (6.34) complete the interpretation of G d as a (d + 1)-points Green function.7. Discussion and conclusionIn this paper, we have given a 2d-conformal field theoretical derivation of the generalizedMacMahon function by using ideas from “transfer matrix method” and q-deformed QFT 2 .Among our results, we mention:(1) The usual vertex operators Γ ± (z) of the bosonic c = 1 conformal field theory appear asthe level one of the following hierarchy,( ∞Γ (p)− (z)|0〉=exp ∑n=1)iz n J −nn(1 − q n ) p−1 |0〉, p 1where q = exp(−g s ). These local operators, which coincide in the limit q → 0; that is when g sgoes to ∞, can be obtained from Γ ± (z) by making the substitutionz n →z n(1 − q n , p 2.) p−1The Γ (p)− s form then an infinite hierarchy of q-deformed vertex operators and obey commutationrelations quite similar to those satisfied by the level one Γ −(1) (z) = Γ −(z). In particular we have,Γ (1)+(7.1)(7.2)(1)Γ(p)−(q) = G p(q)Γ (p) (1)(q)Γ + (1),−where G p (q) is precisely the generalized p-dimension MacMahon function. We also have thefollowing general relation,〈0|Γ + (1) (z 1)Γ − (l+1) (z l )|0〉= ∏ ··· ∏ [ ∏() ] 1(7.4)(1 − q k 0+k 1 +···+k l z l.k l =0 k 1 =0 k 0 =0z 1)This relation can be given an interpretation as l copies of c =∞free CFT 2 representations.Indeed, setting z lz 1= q, q k 0+k 1 +···+k l= Q k q k 0 with Q k = q k 1+···+k landZ 2 (Q k ,q)=∞∏(k 0 =01(1 − Q k q k 0)we can put the right-hand side of (7.4) like,), k = (k 1 ,...,k l ),(7.3)(7.5)