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

L.B. Drissi et al. / Nuclear Physics B 804 [PM] (2008) 307–341 339D 1 : |z 2 | 2 +|z 3 | 2 = t,D 2 : |z 1 | 2 +|z 3 | 2 = t,D 3 : |z 1 | 2 +|z 2 | 2 = t.(A.16)From these equations, we see that each divisor D i is a projective line with Kähler parameter t.The equality of the Kähler parameters t 1 = t 2 = t 3 = t of these projective lines may be also interpretedas due to the permutation symmetry of the {z i } projective coordinate variables of P 2 .Thisfeature translates, in the language of toric geometry, as corresponding to having an equilateraltriangle for the real base B 2 .(b) The divisors {D i } are precisely the ones we get by taking the large complex structure limit(A.4) of the complex curve Eq. (A.12). Under this condition, Eq. (A.12) reduces then to thedominant monomialμz 1 z 2 z 3 = 0.(A.17)Notice that the above relation is obviously invariant under the C ∗ transformations (A.10) sinceμ(λz 1 )(λz 2 )(λz 3 ) = λ 3 (μz 1 z 2 z 3 ) = 0.(A.18)To have more insight about the elliptic curve E (t,∞) with large complex structure; |μ|→∞,itis interesting to solve Eq. (A.17). There are three solutions classified as follows:(i) z 1 = 0, what ever the two other complex variables z 2 and z 3 are; provided that(z 2 ,z 3 ) ≠ (0, 0),(z 2 ,z 3 ) ≡ (λz 2 ,λz 3 ).(A.19)But these relations are nothing but the definition of the divisor D 1 in type IIB geometry.(ii) z 2 = 0, what ever the other complex variables z 1 and z 3 are; provided that(z 1 ,z 3 ) ≠ (0, 0),(z 1 ,z 3 ) ≡ (λz 1 ,λz 3 ),(A.20)describing then the divisor D 2 .(iii) z 3 = 0, what ever the other complex variables z 2 and z 1 are; provided that(z 1 ,z 2 ) ≠ (0, 0),(z 1 ,z 2 ) ≡ (λz 1 ,λz 2 ),(A.21)associated with the divisor D 3 .To conclude the boundary (∂P 2 t ) of the projective plane P2 t is indeed described by an ellipticcurve with a Kähler parameter t inherited from the P 2 t one; but with a large complex structureμ; see also footnote 7. The limit μ →∞explains the degeneracy property in the baseΔ 1 (A.3).