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

322 L.B. Drissi et al. / Nuclear Physics B 804 [PM] (2008) 307–341Fig. 8. A typical 4-vertex in O(−3 − m) → WP1,1,1,m 3 using 2d partitions. This is a spatial vertex made of three planar3-vertices: (abc), (def) and (ghi).Δ 1 = triangle ABC,Δ 2 = triangle ABD,Δ 3 = triangle ACD.These triangles are boundary faces of the tetrahedronABCD.Inside of the tetrahedron, the toric fiber isT 3 = S 1 × S 1 × S 1 .(3.50)(3.51)(3.52)On each triangle face, a circle shrinks leaving T 2 .On each edge of a triangle, one more circle shrinks leaving S 1 .At the vertex A, all 1-cycles of T 3 shrinks down to zero.The property captured by Eqs. (3.49) means that we may relate the 4-vertex C ΛΣΥ Γ tothree planar vertices of the triangles (3.50). This can be done by expressing the 3d partitions(Λ,Σ,Υ,Γ)in terms of 2d partitions (a, b, c), (d, e, f), (g, h, i) and (j, k, l) as followsΛ = (a, d, g),Υ = (e, ∅, f),Σ = (b, c, ∅),Γ = (∅, h, i).(3.53)The decomposition (3.53) is illustrated on the formal Fig. 8 where the three triangles are representedin different colors.Substituting (ΛΣΥ Γ ) as in Eq. (3.53), we can first rewrite C ΛΣΥ Γ like C (adh)(bc∅)(e∅f)(∅hi) .The latter reads immediately from Fig. 8 and is given byC (adg)(bc∅)(e∅f)(∅hi) = C abc C def C ghi ,where C abc , C def and C ghi are topological 3-vertices with the explicit expression Eq. (2.4).(3.54)β) The function Z X4 The toric web-diagram of the X 4 4-fold is given by Fig. 9.The corresponding partition function Z X4 can be computed by specifying the 4-vertices, propagators,framings and using Feynman like rules. Notice that in the 2d partition set up, the toricwebs of X 4 and H 3 are as in Fig. 10.Using the momenta prescriptions described by the Young diagrams of the Fig. 10 as wellas trivial boundary conditions for the external legs, the partition function reads in terms of theKähler modulus t of X 4 as follows:Z H3 = ∑ [(Aτν T ωϕ T ρσ T)(B ετ T χρ T ψς T)(F υɛ T ιψ T κω T)(G ϕκ T χσ T ις T)H τυωϕρσεχικψς],{ϰ} (3.55)