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

312 L.B. Drissi et al. / Nuclear Physics B 804 [PM] (2008) 307–341Fig. 2. Resolved conifold O(−1) ⊕ O(−1) → P 1 made of two planar 3-vertices.Fig. 3. Toric web-diagram of O(−3) → P 2 made of three planar vertices.Eq. (2.4) involves the product of skew-Schur functions S μ/η . It reduces, for the closed topologicalstring case, to∞∏ (Z C 3(q) = C ∅∅∅ (q) = 1 − qn ) −nn=1(2.5)which is nothing but the 3d MacMahon function.Example 2 (Resolved conifold X 3 = O(−1) ⊕ O(−1) → P 1 ). The resolved conifold has oneKähler parameter t parameterizing the size of the projective line P 1 . Fig. 2 describes its toricweb-diagram.This local threefold X 3 is obtained by gluing two 3-vertices along one edge leaving then fouropened external legs.In the simplest case where there is no boundary terms on the external legs, the partition functionof the closed topological string on the resolved conifold is given by,∑Z X3 (q, t) = C ∅∅ν (q)(−1) |ν| e −|ν|t C ∅∅ν T (q).(2.6)2d partitions νIn this relation, ν T is the transpose of the 2d partition ν with |ν| boxes and C ∅∅ν (q) is as inEq. (2.4) by setting the boundary conditions as λ =∅and μ =∅.Example 3 (Local P 2 : X 3 = O(−3) → P 2 ). The local P 2 has one Kähler modulus t parameterizingthe size of the projective plane P 2 . Fig. 3 describes its toric web-diagram.