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

L.B. Drissi et al. / Nuclear Physics B 801 [FS] (2008) 316–345 319vertex operators. In Section 4, we derive the generalized MacMahonfunction G n for 4d and 5d using transfer matrix method and q-deformed vertex operators Γ ±(3)and Γ (4) . In Section 5, we give the result for generic d-dimensions. In Section 6, we derive(i) A q-deformed 2d quantum field theoretical proof of the conjectured MacMahon functionG d .(ii) An interpretation of G d using q-deformed c = 1 conformal field theory rather than CFT 2free field theory with central charge c →∞.(iii) For d 4 G d cannot be the generating function of d-generalized partitions; but rather ofa subclass of d-partitions with very specific boundary conditions.The organization of this paper is as follows:In Section 2, we introduce the usual vertex operators Γ ± of the c = 1 2d conformal modeland give some of their properties essential for the next steps. In Section 3, we revisit the CFT 2derivation of 3d-generalized MacMahon function G 3 using transfer matrix method. We also introducethe q-deformed Γ (2)±±O j (x j ) vertex operators involved in G d (q) re-interpreted as (d + 1)-point correlation functionG d+1 (z 0 ,...,z d ) in q-deformed c = 1CFT 2 . In the conclusion section, we summarize the mainresults of the paper accompanied with a discussion. In Appendices A and B, we give more detailson the proofs of identities used in the present study.2. Vertex operators: Useful propertiesIn this section, we explore some basic properties of the vertex operators Γ ± (z) in c = 1 2dconformalfield theory. We study their commutation relations algebra in connection with thecounting of the Hilbert space states and the 2d-partitions (Young diagrams). We also give specialfeatures of Γ ± (z) which has motivated us to look for the relations (1.2)–(1.3).2.1. Vertex operators in c = 1 CFT 2As the c = 1 field vertex operators Γ ± (z), z ∈ C, have been well studied and are quite knownin 2d conformal field theory [29,35,36], we shall come directly to the main points by consideringthe three following materials needed for the study of Γ ± (z) and their extensions to be consideredin this study:(1) U(1) Kac–Moody algebraIn CFT 2 on the complex line C parameterized by the coordinate z, theU(1) Kac–Moodyalgebra is generated by the holomorphic current J(z) obeying the following operator productexpansion (OPE)1J(z 1 )J (z 2 ) = + regular terms.(z 1 − z 2 ) 2(2.1)Using the Laurent expansion,J(z)= ∑ z −n−1 J n ,n∈Z∮ dzJ n =2iπ zn J(z),(2.2)the above OPE algebra reads as follows[J n ,J m ]=nδ n+m,0 .We have, amongst others, (J n ) † = J −n and J n |0〉=0forn 1.(2.3)