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

322 L.B. Drissi et al. / Nuclear Physics B 801 [FS] (2008) 316–345and Γ − (w) are non-commuting operators, we have,〈0|Γ + (z)Γ − (w)|0〉 ≠〈0|Γ − (z)Γ + (w)|0〉.(2.20)We will develop this issue much more later when we come to the derivation of Eqs. (1.2)–(1.3)by using correlation functions.(ii) As Γ − (z) involves all monomials in J −ni ,J λ −n ≡ ∏ i1(J −ni ) λ i,(2.21)where λ = (λ 1 ,λ 2 ,...) is a 2d-partition, the state Γ − (z)|0〉 is reducible and is given by a sumover all possible 2d-partitions λ. In particular we have for z = 1,∑Γ − (1)|0〉= |λ〉.(2.22)2d partitions λA similar relation is also valid for 〈0|Γ + (1). More generally, this relation extends as Γ − (1)|μ〉and involves the Schur function SμSchur (q) [35]. With these tools, we are in position to proceedfor higher dimensional generalizations.3. The 3d-MacMahon function revisitedOur main objectives here are:(i) revisit the derivation of Z 3d ,(ii) use CFT 2 explicit computations to give arguments which support the existence of a hierarchyof level p vertex operators Γ (p)± .To reach this goal, we first give some details on 3d-partitions (known also as plane partitions)and its generating functional Z 3d . Then we present the explicit computation of the function Z 3dusing transfer matrix method. As mentioned in the introduction, Z 3d is precisely the amplitudeof the topological 3-vertex of closed strings on C 3 . There, the q-parameter is given byq = exp(−g s ),(3.1)with g s being the topological string coupling constant. Z 3d is also the partition function of cornermelting 3d-crystals.3.1. Plane partitions and 3d-Hilbert statesTo begin notice that, from the view of combinatory analysis, the 3d-MacMahon function G 3dcan be defined by the following partition function∑Z 3d =q |Π (3)| ,(3.2)3d partitions Π (3)where |Π (3) | is the number of boxes of the 3d-generalized Young diagram. This relation may bealso written as∑ 〈Z 3d =Π(3) ∣ qH ∣ Π(3) 〉 ,(3.3)3d partitions Π (3)