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L.B. Drissi et al. / Nuclear Physics B 804 [PM] (2008) 307–341 333Fig. 12. Toric graph of O(−3) → E (t,∞) . The compact part is E (t,∞) = (∂P 2 ) with the usual three vertices. Its toricfrontier consists of three intersecting P 1 ’s in the homology class of 2-torus. (a) Figure (on left) represents real skeleton.(b) Figure (on right) gives its fattening.and carry the following q i -charges under the U(1) gauge symmetry,(q 0 ,q 1 ,q 2 ,q 3 ,q 4 ,q γ ) = (−m, 1, 1, 1,m,−3),where m is a priori equal to 3; but in general can take any integral value.(6.7)6.2. GeneralizationThe construction we have developed here above can be generalized to other Calabi–Yau manifolds.Below we make a comment on two kinds of generalizations. The first extension deals withthe gauged sigma model realization of local genus g-Riemann surfaces for g 2. The secondgeneralization concerns sigma model approach for higher complex dimensional compact toricCalabi–Yau manifolds.6.2.1. Local genus g-Riemann surfacesSo far we have seen that for each local elliptic curve, it is associated a U(1) gauge symmetry.This gauge symmetry is inherited from the P 2 model. Since local genus g-Riemann surfaces canbe engineered by gluing several local elliptic curves, we conclude that a class of local genusg-Riemann surfaces could be described by higher rank Abelian U n (1) gauged supersymmetricfield model type. The rank n of the gauge symmetry depends on the way the gluing is done.To illustrate the idea, let us give the example of g = 2-Riemann surface described by a 2DU 2 (1) gauged N = 2 supersymmetric sigma model.The local g = 2-Riemann surface in the large complex structures limits can be engineered bygluing two local elliptic curves with compact base E (t,∞)1= ∂P 2 1and E(t,∞)2= ∂P 2 2. In the sigmamodel approach, we distinguish different representations according to whether P 2 1 and P2 2 havean intersection point or edge.In the first case, the sigma model involves five complex field variables,(z 1 ,z 2 ,z 3 ,z 4 ,z 5 )and a U 2 (1) gauge invariance under which these complex variables have the following gaugecharges(q1i)= (1, 1, 1, 0, 0),(6.9)(q2i)= (0, 0, 1, 1, 1).(6.8)
334 L.B. Drissi et al. / Nuclear Physics B 804 [PM] (2008) 307–341The field theoretic equations describing the compact part of the moduli space of supersymmetricvacua are given by,{ {|z1 | 2 +|z 2 | 2 +|z 3 | 2 = t 1 , |z3 | 2 +|z 4 | 2 +|z 5 | 2 = t 2 ,(6.10)z 1 z 2 z 3 = 0,z 3 z 4 z 5 = 0,where t 1 and t 2 are respectively the Kähler moduli of the projective planes P 2 1 and P2 2 .Theholomorphic constraint equations z 1 z 2 z 3 = 0 and z 3 z 4 z 5 = 0 are implemented in the gauged supersymmetricsuperfield model by two chiral superfields Υ 1 and Υ 2 with gauge charges (q 1 γ ,q2 γ )equal to (−3, 0) and (0, −3) respectively.Notice that Eq. (6.10) describes indeed a complex curve with genus g = 2. The toric threefoldbased on this genus g = 2 curve is parameterized by seven complex variables,(z 0 ,z 1 ,z 2 ,z 3 ,z 4 ,z 5 ,z 6 ),(6.11)with gauge charges as(q1i)= (−3, 1, 1, 1, 0, 0, −3),(6.12)(q2i)= (−3, 0, 0, 1, 1, 1, −3).The gauged supersymmetric field theoretical equation−m|z 0 | 2 +|z 1 | 2 +|z 2 | 2 +|z 3 | 2 + m|z 6 | 2 = t 1 , z 1 z 2 z 3 = 0,−n|z 0 | 2 +|z 3 | 2 +|z 4 | 2 +|z 5 | 2 + n|z 6 | 2 = t 2 , z 3 z 4 z 5 = 0,(6.13)where m and n are in general arbitrary integers; but can be set equal to 3 to keep in touch withthe first Chern class of the complex two-dimensional projective plane.These relations involve seven complex variables constrained by four complex constraint equationsleaving then three complex variables free. Note also that the first relation of the above equationdescribes O(m) ⊕ O(−m) → E (t,∞)1while the second describes O(n) ⊕ O(−n) → E (t,∞)2.6.2.2. Higher-dimensional toric CY manifoldsThe gauged supersymmetric sigma model for the boundary surface of local P 2 that we haveconsidered in this paper can be extended for compact divisors of local P n−1 . The latter is givenby the following U(1) gauge invariant complex dimension n hypersurfacen∑n|z 0 | 2 + |z i | 2 = t,(6.14)i=1embedded in C n+1 parameterized by the local coordinates {z 0 ,z 1 ,z 2 ,...,z n } with gauge charge(q 0 ,q 1 ,q 2 ,...,q n ) = (−n, 1, 1,...,1).(6.15)In Eq. (6.14), t is the usual Kähler parameter of P n−1 . To describe the compact (divisor) boundary∂(P n−1 ) of the toric n-fold, we supplement the hypersurface equation by the following extragauge covariant constraint relation,n∏z i = 0.(6.16)i=1
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UNIVERSITÉ MOHAMMED V - AGDALFACUL
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TABLE DES MATIÈRES3.3 Invariants t
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TABLE DES MATIÈRES8.2 Fonctions de
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Avant ProposCe travail à été eff
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Lab/UFR PHEUne thèse représente u
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10Lab/UFR PHE
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Contributions à l’Etude du Verte
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2.1 Généralités sur les variét
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2.2 Conifoldavec (y 1 , y 2 ) ≠ (
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2.2 Conifoldoù µ est un nombre co
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2.3 Variétés de CY toriquesFig. 2
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2.3 Variétés de CY toriquesEn gé
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2.3 Variétés de CY toriquessur L
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2.3 Variétés de CY toriquesz3z1z2
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2.3 Variétés de CY toriquesFig. 2
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du vertex U 3r α = |z 1 | 2 − |z
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3.1 Théorie des cordes topologique
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3.2 Dualité corde ouverte / corde
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3.3 Invariants topologiquesNotons a
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3.3 Invariants topologiquesDans le
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4.2 Fonction de partition perpendic
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4.3 Version raffinée de la fonctio
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4.3 Version raffinée de la fonctio
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4.4 Modèle du cristal fondu et con
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4.4 Modèle du cristal fondu et con
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4.5 Invariants topologiques dans le
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4.6 Contribution : Generalized MacM
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(Γ + (z) = exp −i ∑ )1n z−n
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Fonctions de Schur et MacMahonFig.
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Bibliographie[1] J. Polchinski, Str
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BIBLIOGRAPHIE[27] A.A. Belavin, A.
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BIBLIOGRAPHIE[57] A. Braverman and
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BIBLIOGRAPHIE[87] Yukiko Konishi, I
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BIBLIOGRAPHIE[119] D. A. Cox, The H
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BIBLIOGRAPHIE[150] C. Weiss and M.
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BIBLIOGRAPHIE[180] H. Awata and H.
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UNIVERSITÉ MOHAMMED V - AGDALFACUL
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