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

L.B. Drissi et al. / Nuclear Physics B 804 [PM] (2008) 307–341 327Fig. 11. (a) Left: Toric graph of P 2 . (b) Right: Toric graph of E (t,∞) where we have added a hole to avoid confusion.From the toric diagram of P 2 , one also see that E (t,∞) describes indeed a toric complex onedimensionalcurve defining the toric boundary of P 2 ; i.e.:E (t,∞) ≡ ∂P 2 .It is this curve that will be used to deal with the local 2-torus in the large complex structure limit.5.1.2. Elliptic curve E (t,∞)An interesting question concerns the derivation of the defining equation describing the ellipticcurve E (t,∞) . From the above analysis, it is not difficult to see that E (t,∞) is given by thefollowing system of equations (see Fig. 11),⎧⎨ |z 1 | 2 +|z 2 | 2 +|z 3 | 2 = t,z (5.8)⎩ i ≡ z i e iqiα ,z 1 z 2 z 3 = 0.In these relations, we have three complex variables (z 1 ,z 2 ,z 3 ) subject to three constraint equations.The two first equations, which are real, are just the defining linear sigma model equationof P 2 . They will be interpreted as the field equation of motion of the auxiliary D-field in supersymmetricsigma model.The third relation, which is covariant under U(1) gauge symmetry, is an extra complex conditionimplemented in order to restrict P 2 geometry to its toric boundary ∂P 2 . It will be interpretedlater as the equation of motion of the auxiliary F-fields.Notice that, the implementation of the boundary condition is a new feature. It can be thenviewed as:(i) a generalization of the usual approach for dealing with sigma model realization of toricmanifolds,(ii) a way to approach genus g Riemann surfaces,(iii) a method that can be used to describe the toric boundary of more general complex n-dimensionaltoric Calabi–Yau manifolds. We will make a comment regarding this point in theconclusion section.Notice finally that for t ≠ 0 the three complex variables cannot vanish simultaneously, i.e.,(5.7)(z 1 ,z 2 ,z 3 ) ≠ (0, 0, 0).(5.9)