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

316 L.B. Drissi et al. / Nuclear Physics B 804 [PM] (2008) 307–341Fig. 7. On left, the real skeleton of the toric web-diagram of the fibration of O(−3) → E (t,∞) . It could be interpreted asa triangulation of the cylinder of base E (t,μ) , given by the boundary of a triangle, and a non-compact line as a fiber. Onright, the usual cylinder R × S 1 .The toric web-diagram of Y 2 is non-planar and can be thought of as the triangulation of thetopological cap of [28]. Below, we will refer to Y 2 as the “toric cap”. Notice also the followingfeatures:(a) The complex surface Y 2 is a compact divisor of H 3 ; it is made of the union of three complexprojective planes, which we denote as P 2 1 , P2 2 and P2 3 .The projective planes P 2 i belong to three different C3 spaces of the ambient C 4 .(b) The toric web-diagram of Y 2 is made of three triangles as shown 4 in Fig. 6. Recall that thetoric web-diagram of a generic projective plane is given by a triangle (Fig. 3).The projective planes (triangles) have mutual intersections I ij along complex projective lines(edges) and a non-planar tri-intersection vertex I 123 .(2) Toric cylinderThis divisor corresponds to think about H 3 as the fibration O(+3) → Ỹ 2 .Here the base Ỹ 2 is a non-compact complex surface given byỸ 2 = O(−3) → E (t,∞) .(3.14)Its first Chern class c 1 (T ∗ Ỹ 2 ) is equal to −3. Here also the toric web-diagram of E (t,∞) is theboundary of the triangle and O(−3) is a non-compact line. The toric web-diagram of Ỹ 2 is nonplanar;it can be thought of as the triangulation of the cylinder R × S 1 . We will refer to Ỹ 2 as the“toric cylinder” whose toric web-diagram is shown in Fig. 7.The Ỹ 2 divisor of H 3 is made of the union of three sheets O(−3) → P 1 ibelonging to threedifferent C 3 spaces of the ambient C 4 .From Eqs. (3.13) and (3.14), it is clear that the elliptic curve E (t,∞) is an intersecting curveof the complex surfaces Y 2 and Ỹ 2 .3.3. Topological partition functionTo build the non-planar 3-vertex formalism for the local elliptic curve H 3 , we will follow theconstruction used in the derivation of the usual topological 3-vertex method [28].4 Y 2 can be imagined as the triangulation of the cap.