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

Nuclear Physics B 804 [PM] (2008) 307–341www.elsevier.com/locate/nuclphysbNon-planar topological 3-vertex formalismLalla Btissam Drissi a,b , Houda Jehjouh a,b , El Hassan Saidi a,b,∗a Lab/UFR – Physique des Hautes Energies, Faculté des Sciences, Rabat, Moroccob GNPHE, Groupement National de Physique des Hautes Energies, Siège focal: Faculté des Sciences, Rabat, MoroccoReceived 4 January 2008; accepted 15 April 2008Available online 30 April 2008AbstractUsing embeddings of complex curves in the complex projective plane P 2 ,wedevelopanon-planar topological3-vertex formalism for topological strings on the family of local Calabi–Yau threefolds X (m,−m,0) =O(m) ⊕ O(−m) → E (t,∞) . The base E (t,∞) stands for the degenerate elliptic curve with Kähler parametert; but a large complex structure μ;i.e.|μ|→∞. We also give first results regarding A-model topologicalstring amplitudes on X (m,−m,0) . The 2D U(1) gauged N = 2 supersymmetric sigma models of the degenerateelliptic curve E (t,∞) as well as for the family X (m,−m,0) are studied and the role of D- and F-termsis explicitly exhibited.© 2008 Elsevier B.V. All rights reserved.Keywords: Topological string; Topological vertex; Hypersurfaces in the local projective plane; Supersymmetric linearsigma models with D- and F-terms1. IntroductionTopological string theory [1–3] is a powerful method to deal with the 4D N = 2 supergravityPlanck limit of the compactification of type II superstring on Calabi–Yau (CY) threefolds X 3[4,5]. The study of local Gromov–Witten theory of curves in non-compact Calabi–Yau threefolds[6–10] and the OSV conjecture [11], relating microscopic 4D black holes to 2D q-deformedYang–Mills theory [12–18] and [41–43], have given an additional impulse to the revival interestin the study of the topological field and string theories [19–26] and [27]. Several important resultshave been obtained in the few last years; in particular the development of the topological tri-* Corresponding author at: Lab/UFR – Physique des Hautes Energies, Faculté des Sciences, Rabat, Morocco.E-mail addresses: drissilb@gmail.com (L.B. Drissi), jehjouh@gmail.com (H. Jehjouh), h-saidi@fsr.ac.ma(E.H. Saidi).0550-3213/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysb.2008.04.025