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

332 L.B. Drissi et al. / Nuclear Physics B 804 [PM] (2008) 307–341The upper index of z (i)jrefers to the corresponding chart U i . Note that on each chart, we have therelationz (i)1 z(i) 2 z(i) 3= 0, i = 1, 2, 3.(5.34)The patches U i could be interpreted as the three “toric pants” needed for building E (t,∞) andmay be related to the topological pant considered in [30]. By gluing these three patches, onereproduces E (t,∞) .6. Local 2-torusThe local complex surface X 2 = O(−m) → E (t,∞) we have considered so far is not a Calabi–Yau 2-fold. The first Chern class c 1 (T ∗ X 2 ) of this variety is equal to −m. For our concern, thissurface is thought of as a divisor of the local Calabi–Yau threefold,O(m) ⊕ O(−m) → E (t,∞) .(6.1)6.1. Field modelThe supersymmetric field model relations describing the toric Calabi–Yau threefold (6.1) canbe easily derived from previous study. They are given by the following system of componentfield equations,{|z1 | 2 +|z 2 | 2 +|z 3 | 2 + 3|z 4 | 2 − 3|z 0 | 2 = t,(6.2)z 1 z 2 z 3 = 0.Here z 0 and z 4 parameterize the non-compact directions and (z 1 ,z 2 ,z 3 ) are as before. The toricgraph of this local threefold is shown on Fig. 12; it has three tetra-valent vertices.To get the superfield Lagrangian density L locT 2, we think about Eqs. (6.2) as the field equationsof motion of the D and F i auxiliary fields.The first relation is associated withδL locEδD = 0,while the second follows from,δL locE= 0.δF iThe result is∫ 3∑∫L locE = d 4 θ ¯Φ i e 2V Φ i +i=1∫+ L gauge (V ) − 2td 4 θ ( ¯Φ 0 e −2mV Φ 0 + ¯Φ 4 e 2mV Φ 4)( ∫d 4 θV + g)d 2 θΦ 1 Φ 2 Φ 3 Υ + hc ,(6.3)(6.4)(6.5)where Υ is a Lagrange superfield multiplier capturing the constraint restricting the field variablesto the boundary of P 2 .In addition to the U(1) gauge multiplet V , the chiral superfields of this model are(Φ 0 ,Φ 1 ,Φ 2 ,Φ 4 ,Φ 5 ,Υ)(6.6)