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

320 L.B. Drissi et al. / Nuclear Physics B 804 [PM] (2008) 307–341with i = 1,...,N 1 ; a = 1,...,N 2 and α = 1,...,N 3 ,(ii) the product of three kinds of 2d partitionsP = λ ⊗ μ ⊗ ν, (P iaα ) = (λ i μ a ν α ).(3.36)The boundary ∂P of a generic 4d partition P can be defined in two ways. First in terms of 3dpartitions as follows∂P = (Λ,Π,Σ,Υ).We also have the following particular boundary conditionscase I:case II:case III:case IV:∂P = (∅,Π,Σ,Υ),∂P = (∅, ∅,Σ,Υ),∂P = (∅, ∅, ∅,Υ),∂P = (∅, ∅, ∅, ∅),where ∅ stands for the “3d vacuum” (no boundary condition).Second by using 2d partitions to define boundary of P like(3.37)(3.38)∂P = ( [a, b, c]; [d, e, f]; [g, h, i]; [j, k, l] ) ,where [a, b, c],...and [j, k, l] stand for the boundaries of the 3d partitions Λ,...and Υ .Notice that the second representation is more richer since along with the configuration(3.39)∅ =[∅, ∅, ∅],we have moreover the two following extra configurations(3.40)[∅, b, c],[∅, ∅, c].For the case I of Eq. (3.38) correspond then the three following boundary configurations⎧⎪⎨case i: ([∅, b, c]; [d, e, f]; [g, h, i]; [j, k, l])∂P = case ii: ([∅, ∅, c]; [d, e, f]; [g, h, i]; [j, k, l]),⎪⎩case iii: ([∅, ∅, ∅]; [d, e, f]; [g, h, i]; [j, k, l]) ∣(3.41)(3.42)where the last one (case iii) is the case I described by the first relation of Eq. (3.38).This property indicates that one disposes of different ways to deal with 4d partitions eitherthe simplest one using 3d-partitions or the more refined on involving 2d partitions. Below weconsider both representations.Notice moreover that given a 4d partition P, we can associate to it various kinds of transposepartitions. Using the particular realization Eq. (3.36), the corresponding transposes read asλ T ⊗ μ ⊗ ν, λ ⊗ μ T ⊗ ν, λ ⊗ μ ⊗ ν T ,λ T ⊗ μ T ⊗ ν, λ ⊗ μ T ⊗ ν T , λ T ⊗ μ ⊗ ν T , λ T ⊗ μ T ⊗ ν T ,(3.43)where λ T stands for the usual transpose of the Young diagram λ.The exact mathematical definitions and the full properties of 4d partitions are not our immediateobjective here; they need by themselves a separate study. Here above we have given just theneeded properties to set up the structure of the 4-vertex formalism and its restricted non-planar3-vertex method.