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

336 L.B. Drissi et al. / Nuclear Physics B 804 [PM] (2008) 307–341We have also developed the gauged supersymmetric sigma model realization that underliesthe geometry the local 2-torus with |μ|→∞and exhibited explicitly the role of D- and F-terms.We have discussed as well how this construction could be extended to local genus g-Riemannsurfaces O(m) ⊕ O(2 − 2g − m) → Σ g in the limit of large complex structures.The results obtained in the field theory part of the paper may also be viewed as an explicitanalysis regarding implementation of F-terms in the Witten’s original work on phases of N = 2supersymmetric theories in two dimensions [47].AcknowledgementsThe authors thank the International Centre for Theoretical Physics, and S. Randjabar Daemifor kind hospitality at ICTP. This research work is supported by Protars III CNRST-D12/25.Appendix AIn this appendix, we give useful properties on the complex projective plane and on particularaspects on the complex curves in P 2 .More precisely denoting by P 2 t , the projective plane with Kähler parameter t and by E(t,μ)the following elliptic curve in P 2 t ,E (t,μ) : z 3 1 + z3 2 + z3 3 + μz 1z 2 z 3 = 0we want to show that the boundary ∂(P 2 t ) is nothing but the degenerate limit μ →∞of E(t,μ) ;that is∂ ( P 2 )t ≃ E (t,∞) .(A.1)This question can be also rephrased in other words by using the fibration,P 2 = B 2 × T 2 ,(A.2)where B 2 is real 2-dimensional base (an equilateral triangle). The boundary ∂(P 2 ) is a toricsubmanifold with fibration∂ ( P 2) = Δ 1 × S 1 ,(A.3)where Δ 1 = (∂B 2 ) is the boundary of a triangle.Clearly, thought not exactly the standard 2-torus S 1 × S 1 , the boundary ∂(P 2 ) has somethingto do with it. It is the large complex structure μ of the elliptic curve E (t,μ) ;say|μ|→+∞.(A.4)As we need both Kähler and complex structures to answer the question (A.1), let us first givesome useful details and then turn to derive the identity ∂(P 2 t ) ≃ E(t,∞) .Projective plane P 2There are different ways to deal the complex projective plane P 2 . Below, we give two dualdescriptions by using the so-called type IIA and type IIB geometries [51].Type IIA geometryIn this set up, known also as toric geometry, the projective plane P 2 is defined by the followingreal 4-dimensional compact hypersurface in C 3 ,|z 1 | 2 +|z 2 | 2 +|z 3 | 2 = t,(A.5)