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

L.B. Drissi et al. / Nuclear Physics B 804 [PM] (2008) 307–341 315Fig. 5. Non-planar toric web-diagram of O(+3) ⊕ O(−3) → E (t,∞) . This is a toric CY3 divisor of the four dimensioncomplex Kähler manifold O(+3) ⊕ O(−3) → P 2 . The hollow triangle ABC refers to the degenerate elliptic curveE (t,∞) . The full triangles ABD, ZCD, BCD refer to the three other projective planes.Fig. 6. Toric web-diagram of O(+3) → E (t,∞) . This figure looks like a “toric cap” obtained by gluing three triangles asshown on the figure.where WP1,1,1,3 3 stands for the complex 3 dimension weighted projective space. To keep in touchwith the Calabi–Yau condition, we promote Y 4 to the toric Calabi–Yau 4-foldX 4 = O(−6) → WP1,1,1,3 3 ,and in general toX 4 = O(−3 − m) → WP1,1,1,m3with m 1.(3.11)(3.12)3.2.2. Toric cap and toric cylinderFrom Eq. (3.8), one distinguishes two special divisors of the local degenerate ellipticcurve H 3 :(1) “Toric cap”: see Fig. 6This divisor corresponds to H 3 taken as the fibration O(−3) → Y 2 . The base Y 2 is a compactcomplex surface given byY 2 = O(+3) → E (t,∞) .(3.13)The toric web-diagram of the complex surface Y 2 is exhibited in Fig. 5.ThefirstChernclassc 1 (T ∗ Y 2 ) is equal to +3. The toric web-diagram of E (t,∞) is given bythe boundary of the triangle and O(+3) is a compact line.