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

330 L.B. Drissi et al. / Nuclear Physics B 804 [PM] (2008) 307–3415.3.1. Moduli space of vacuaIn the supersymmetric vacuum, the vanishing condition of the scalar potential V = V(z) ofthe model (5.13) reads as|D| 2 +|F 0 | 2 +|F 1 | 2 +|F 2 | 2 +|F 3 | 2 +|F γ | 2 = 0.(5.22)The dependence of the scalar potential V in the scalar fields z is obtained by replacing theauxiliary fields D and F i by their explicit expressions in terms of the matter fieldsD = D(z 0 ,z 1 ,z 2 ,z 3 ,z γ ),F i = F i (z 0 ,z 1 ,z 2 ,z 3 ,z γ ).(5.23)These expressions are obtained by using the equations of motionδLδD = 0,δLδ ¯F i= 0.Eq. (5.22) is solved as follows,D = 0, F i = 0.(5.24)(5.25)As noted before, D = 0 leads to|z 1 | 2 +|z 2 | 2 +|z 3 | 2 − m|z 0 | 2 = t(5.26)and describes local P 2 for the particular case m = 3.F i = 0 involves five terms: F 0 which is trivial, and the remaining F γ , F 3 , F 2 , and F 1 lead to:z 1 z 2 z 3 = 0,z 1 z 2 z γ = 0,z 3 z 1 z γ = 0,z 2 z 3 z γ = 0,(5.27)where z γ stands for the lowest component field of the chiral superfield Υ . There are severalsolutions of these relations. These solutions may be classified into two sets:(1) The first set is given byz γ = 0.(5.28)Consequently Eqs. (5.27) reduce to the first equation z 1 z 2 z 3 = 0.The moduli background associated with this solution describes exactly the complex surfaceO(−3) → E (t,∞) .(2) The second set corresponds toz γ ≠ 0(5.29)and two variables amongst the three z 1 , z 2 and z 3 vanish.