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

324 L.B. Drissi et al. / Nuclear Physics B 804 [PM] (2008) 307–3413.3.3. Partition function for the local 2-torusThe partition function Z H3 of the local elliptic curve may be obtained by implementing inZ X4 the constraint relations (3.24) capturing the projection X 4 → H 3 .Choosing trivial boundary conditions for the external legs and using:(i) the expression of the 4-vertex (3.54),(ii) the rules of the planar vertex formalism of [28],we can write down directly the expression of the partition function Z H3 . We find∑ (Z H3 =A∗ωϕ T ρσ T B ∗ χρ T ψς T F ∗ ιψ T κω T G ∗ ϕκ T χσ T ις T Hωϕρσχικψς) ∗ ,withξ,ρ,σ,η,υ,τ,ς,θA ∗ ωϕ T ρσ T = C ∅ωϕ TC ∅ρσ T,B ∗ χρ T ψς T = C ∅χρ TC ∅ψς T,F ∗ ιψ T κω T = C ∅ιψ TC ∅κω T,G ∗ ϕκ T χσ T ις T = C ∅ϕκ TC ∅χσ TC ∅ις T,(3.58)(3.59)together withH ∗ ωϕρσχικψς =∏μ={ω,ϕ,ρ,σ,χ,ι,κ,ψ,ς}(−e−t ) |μ| q κ(μ) ,(3.60)where the factors C αβγ areasinEq.(2.4).In what follows, we turn to study the field theory set up of the local 2-torus by starting bylocal P 2 model.4. Sigma model for local P 2In this section, we first review briefly the supersymmetric sigma linear model realization ofthe local P 2 model. This model is useful for the purpose of this paper.We also use this field realization to fix convention notations and to introduce some mathematicalobjects and their physical interpretations.The local P 2 model is nicely formulated in the language of 4D, N = 1 supersymmetry whichis, roughly, equivalent to the usual 2D, N = 2 supersymmetry. The complex two dimensionprojective plane P 2 has one Kähler parameter t, interpreted as the Fayet–Iliopoulos (FI) couplingconstant in the supersymmetric gauge theory.The U(1) gauged linear sigma theory describing the local P 2 target space geometry involvesthe following 4D, N = 1 superfields (supersymmetric representations):(1) A U(1) gauge superfield V = V(x,θ,¯θ) which reads, in the Wess–Zumino gauge, as follows:V =−θσ μ ¯θA μ − i ¯θ 2 θλ+ iθ 2 ¯θ ¯λ + 1 2 θ 2 ¯θ 2 D,(4.1)where (x μ ,θ a , ¯θȧ) stands for the 4D, N = 1 superspace coordinates.