344 L.B. Drissi et al. / Nuclear Physics B 801 [FS] (2008) 316–345where p j are giv<strong>en</strong> byp j =(p − 1)!, j = 0,...,p− 1.j!(p − j − 1)!(A.37)(ii) Using the decomposition for Γ (p)− (z),Γ (p)− (z) = O 0(x 0 )O 1 (x 1 )O 2 (x 2 ) ···O p−1 (x p−1 )(A.38)we getO j+1 (x j+1 ) =∏p ji=1Γ (j+1) (− q j z ) , j = 0,...,p− 1.(A.39)App<strong>en</strong>dix B. Combinatorial Eq. (5.17)Here we want to derive the id<strong>en</strong>tity (5.17) namely,s∑k=1C p−2k+p−3 = Cp−1 s+p−2, p 2.This is a standard combinatorial id<strong>en</strong>tity; its proof follows from basic property [37],C k n+1 = Ck−1 n + C k n .(B.1)(B.2)Applying this id<strong>en</strong>tity to Cn k and putting it back into the above relation, we get,(B.3)Cn+1 k = Ck−1 n + C k−1n−1 + Ck n−1 .By in<strong>du</strong>ction, it results,C k n+1 =n∑j=k−1C k−1j.Setting k = p − 1 and n = s + p − 3, we recover the id<strong>en</strong>tity (B.1).(B.4)Refer<strong>en</strong>ces[1] H.N. Temperley, Statistical mechanics and the partition of numbers I: The transition to liquid helium, Proc. R. Soc.London A 199 (1949) 361;H.N. Temperley, Statistical mechanics and the partition of numbers II: The form of crystal surfaces, Proc. CambridgePhilos. Soc. 48 (1952) 683.[2] E.J. van R<strong>en</strong>sburg, The Statistical Mechanics of Interacting Walks, Polygons, Animals and Vesicles, Oxford Univ.Press, Oxford, 2000.[3] C. Weiss, M. Holthaus, From number theory to statistical mechanics: Bose–Einstein cond<strong>en</strong>sation in isolated traps,Chaos Solitons Fractals 10 (1999) 795.[4] V. Elser, Solution of the dimer problem on a hexagonal lattice with boundary, J. Phys. A 17 (1984) 1509.[5] A. Okounkov, N. Reshetikhin, C. Vafa, Quantum Calabi–Yau and classical crystals, hep-th/0309208.[6] R. K<strong>en</strong>yon, An intro<strong>du</strong>ction to the dimer model, math.CO/0310326.[7] R. K<strong>en</strong>yon, A. Okounkov, S. Sheffield, Dimers and amoebae, math-ph/0311005.[8] D. Ghoshal, C. Vafa, c = 1 string as the topological theory of the conifold, Nucl. Phys. B 453 (1995) 121, hepth/9506122.[9] E. Witt<strong>en</strong>, Ground ring of two-dim<strong>en</strong>sional string theory, Nucl. Phys. B 373 (1992) 187, hep-th/9108004.
L.B. Drissi et al. / Nuclear Physics B 801 [FS] (2008) 316–345 345[10] E.H. Saidi, M.B. Sedra, Topological string in harmonic space and correlation functions in S 3 stringy cosmology,Nucl. Phys. B 748 (2006) 380–457, hep-th/0604204;E.H. Saidi, Topological SL(2) gauge theory on conifold, hep-th/0601020.[11] M.R. Douglas, G. Moore, D-branes, Quivers, and ALE instantons, hep-th/9603167.[12] R. Ahl Laamara, M. Ait B<strong>en</strong> Haddou, A. Belhaj, L.B. Drissi, E.H. Saidi, RG Cascades in hyperbolic quiver gaugetheories, Nucl. Phys. B 702 (2004) 163–188, hep-th/0405222.[13] M. Ait B<strong>en</strong> Haddou, A. Belhaj, E.H. Saidi, Geometric <strong>en</strong>gineering of N = 2 CFT_{4}s based on indefinite singularities:Hyperbolic case, Nucl. Phys. B 674 (2003) 593–614, hep-th/0307244.[14] M.A. B<strong>en</strong>haddou, E.H. Saidi, Explicit analysis of Kähler deformations in 4D N = 1 supersymmetric quiver theories,Phys. Lett. B 575 (2003) 100–110, hep-th/0307103.[15] J.J. Heckman, C. Vafa, Crystal melting and black holes, hep-th/0610005.[16] M. Aganagic, A. Klemm, M. Marino, C. Vafa, The topological vertex, Commun. Math. Phys. 254 (2005) 425–478,hep-th/0305132.[17] A. Iqbal, N. Nekrasov, A. Okounkov, C. Vafa, Quantum foam and topological strings, hep-th/0312022.[18] A. Iqbal, C. Kozcaz, C. Vafa, The refined topological vertex, hep-th/0701156.[19] L.B. Drissi, H. Jehjouh, E.H. Saidi, Topological strings on local elliptic curve and non-planar 3-vertex formalism,arXiv: 0712.4249 [hep-th], Lab/UFR-PHE, 0701.[20] V.G. Kac, A.K. Raina, Repres<strong>en</strong>tations of Infinite Dim<strong>en</strong>sional Lie Algebras, Bombay Lectures on Highest Weight,World Sci<strong>en</strong>tific, 1987.[21] V.G. Kac, Infinite Dim<strong>en</strong>sional Lie Algebras, third ed., Cambridge Univ. Press, 1990.[22] E.H. Saidi, M. Zakkari, The G<strong>en</strong>erating functional of higher conformal spin ext<strong>en</strong>sions of the Virasoro algebra,Phys. Lett. B 281 (1992) 67–71.[23] H. Awata, M. Fukuma, Y. Matsuo, S. Odake, Repres<strong>en</strong>tation theory of the W 1+∞ algebra, Prog. Theor. Phys.Suppl. 118 (1995) 343–374, hep-th/9408158.[24] E. Fr<strong>en</strong>kel, V. Kac, A. Ra<strong>du</strong>l, W.-Q. Wang, W 1+∞ and W(gl N ) with c<strong>en</strong>tral charge N, Commun. Math. Phys. 170(1995) 337–358, hep-th/9405121.[25] V.G. Kac, A.K. Raina, Bombay Lectures on Highest Weight Repres<strong>en</strong>tations of Infinite Dim<strong>en</strong>sional Lie Algebras,World Sci<strong>en</strong>tific, 1987.[26] M. Ait B<strong>en</strong> Haddou, Thèse de doctorat de 3ème cycle, Sur Les Repres<strong>en</strong>atations des Algèbres de Lie de dim<strong>en</strong>sionsInfinies, Faculté des Sci<strong>en</strong>ces, Rabat, 1994.[27] H. Awata, M. Fukuma, Y. Matsuo, S. Odake, Determinant formulae of quasi-finite repres<strong>en</strong>tation of W 1+∞ algebraat lower levels, Phys. Lett. B 332 (1994) 336–344, hep-th/9402001.[28] V. Muston<strong>en</strong>, R. Rajesh, Numerical estimation of the asymptotic behaviour of solid partitions of an integer, J. Phys.A 36 (2003) 6651, cond-mat/0303607.[29] A. Okounkov, N. Reshetikhin, Random skew plane partitions and the Pearcey process, math/0503508.[30] C. Vafa, Two-dim<strong>en</strong>sional Yang–Mills, black holes and topological strings, hep-th/0406058.[31] M. Aganagic, H. Ooguri, N. Saulina, C. Vafa, Black holes, q-deformed 2d Yang–Mills, and non-perturbative topologicalstrings, Nucl. Phys. B 715 (2005) 304–348, hep-th/0411280.[32] R. Ahl Laamara, A. Belhaj, L.B. Drissi, E.H. Saidi, Black holes in type IIA string on Calabi–Yau threefoldswith affine ADE geometries and q-deformed 2d quiver gauge theories, Nucl. Phys. B 776 (2007) 287–326, hepth/0611289.[33] O. Foda, M. Wheeler, BKP plane partitions, JHEP 0701 (2007) 075, math-ph/0612018.[34] P. Ginsparg, G. Moore, Lectures on 2D gravity and 2D string theory (TASI 1992), hep-th/9304011.[35] A. Okounkov, N. Reshetikhin, Correlation function of Schur process with application to local geometry of a random3-dim<strong>en</strong>sional Young diagram, math.CO/0107056.[36] N. Saulina, C. Vafa, D-branes as defects in the Calabi–Yau crystal, hep-th/0404246.[37] I.S. Gradshteyn, I.M. Rhyzik, Tables of Integrals, Series and Pro<strong>du</strong>cts, sev<strong>en</strong>th ed., Academic Press, 1980.
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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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Contributions à l’Etude du Verte
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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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du vertex U 3r α = |z 1 | 2 − |z
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Contributions à l’Etude du Verte
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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 topologiquesDans le
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3.3 Invariants topologiquesL’acti
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Chapitre 6Vertex Topologique Raffin
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5.1 Raffinement du vertex topologiq
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5.2 Fonctions de partitions du vert
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Conclusion et perspectivesde ce doc
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Fonctions de Schur et MacMahonayant
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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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