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

L.B. Drissi et al. / Nuclear Physics B 804 [PM] (2008) 307–341 331So Eqs. (5.26) becomez 2 = z 3 = 0: |z 1 | 2 − m|z 0 | 2 = t,z 1 = z 3 = 0: |z 2 | 2 − m|z 0 | 2 = t,z 1 = z 2 = 0: |z 3 | 2 − m|z 0 | 2 = t.(5.30)5.3.2. Case z γ = 0Let us now consider the interesting case z γ = 0 and study the solution of constraint equationz 1 z 2 z 3 = 0. Here also there are several solutions which we list below:(1) Case z 3 = 0but(z 1 ,z 2 ) ≠ (0, 0)In this case the geometry reduces to|z 1 | 2 +|z 2 | 2 − 3|z 0 | 2 = t.(5.31)It describes the local complex projective lineO(−3) → P 1 ,(5.32)which we denote as O(−3) → P 1 3 where the sub-index 3 on P1 3 refers to z 3 = 0.The same thing is valid for z 1 = 0 and z 2 = 0.They describe respectively the local surfaces O(−3) → P 1 1 and O(−3) → P1 2 .(2) Case z 1 = 0, z 2 = 0, z 3 = √ tThis solution describes one of the three possible vertices of O(−3) → E (t,∞) ; the two othervertices are associated with the points:(i) z 1 = 0, z 2 = √ t, z 3 = 0 and,(ii) z 1 = √ t, z 2 = 0, z 3 = 0.(3) Case z 1 = z 2 = z 3 = 0isaP 2 singularityThis solution corresponds to the limit t = 0 where both E (t,∞) and so P 2 collapse down to apoint.The above analysis can be viewed as an interesting step towards the study of topologicalvertex of local genus g-Riemann surfaces (in particular g = 1) by using toric diagrams based onthe curve E (t,∞) . To that purpose, one first has to build the toric realization of basic objects ofthe topological vertex method. For instance, the complex coordinates associated with the verticesof the elliptic curve E (t,∞) are given by the local patchesU 1 : z (1)0 , z(1) 1 , z(1) 2 , z(1) 3= 0,U 2 : z (2)0 , z(2) 1 , z(2) 3 , z(2) 2= 0,U 3 : z (3)0 , z(3) 2 , z(3) 3 , z(3) 1= 0.(5.33)