Contributions Ã l'Etude du Vertex Topologique en ThÃ©orie ... - Toubkal

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JOURNAL OF MATHEMATICAL PHYSICS 50, 013509 2009Refining the shifted topological vertexL. B. Drissi, 1,2,a H. Jehjouh, 1,2,b and E. H. Saidi 1,2,c1 Faculté des Sciences, Laboratory/UFR-Physique des Hautes Energies,Rabat, 1014 Morocco2 Groupement National de Physique des Hautes Energies (GNPHE), Siège focal:FS,Rabat, 1014 MoroccoReceived 29 April 2008; accepted 6 November 2008; published online 7 January 2009We study aspects of the refining and shifting properties of the 3d MacMahonfunction C 3 q used in topological string theory and BKP hierarchy. We derive theexplicit expressions of the shifted topological vertex S q and its refined versionT q,t. These vertices complete results in literature. © 2009 American Instituteof Physics. DOI: 10.1063/1.3040186I. INTRODUCTIONIn the past few years, there has been some interest in the study of the topological vertexformalism of toric Calabi–Yau threefold CY3. 1 This interest has followed the basic result, accordingto which the topological vertex C is a powerful tool to compute toric CY3 topologicalstring amplitudes. 1–4,23 It also came from the remarkable relation between C and the Gromov–Witten invariants of genus g-curves in toric CY3s. 5–7Recently it has been shown that the standard topological three-vertex C q may have twokinds of generalizations: one known as the refining of C q and the other as its shifting.In the first case, the refined topological vertex R q,t is a two parameter function computingthe refined topological string amplitudes of toric CY3s. 8–10 It has been found also thatR q,t computes as well the Nekrasov’s 11 instantons of the topological string free energyFX I , 1 , 2 of four dimensional SUN gauge theories. 12 In Nekrasov’s extension, the usualtopological string coupling constant g top =ln1/q gets replaced by the pair of parameters 1 and 2 . 13 In the second case, the standard MacMahon function C 3 q Refs. 3 and 14 has been extendedto the so-called shifted partition function S 3 q. This is the generating function of the shifted planepartitions and it is used in the study of BKP hierarchy. 15–18The aim of this paper is to contribute to this matter by combining both the refining and theshifting operations to get the refined-shifted topological vertex T q,t extending R q,t andS 3 q obtained recently in literature. More precisely, we want to complete the missing relationspresented in the two following tables:iFirst, we determine the refined version of the shifted MacMahon function S 3 q obtainedby Foda and Wheeler. The refined version of S 3 q, denoted below as T 3 q,t, is missing.It is a two parameter function generating shifted 3d-partitions needed to complete the table,C 3 q = known: Eq. 2.4refining→shifting ↓ ↓ shiftingS 3 q = known: Eq. 2.9refining→R 3 q,t = known: Eq. 2.8T 3 q,t = ... ?. 1.1a Electronic mail: drissilb@gmail.com.b Electronic mail: jehjouh@gmail.com.c Electronic mail: h-saidi@fsr.ac.ma.0022-2488/2009/501/013509/12/$25.0050, 013509-1© 2009 American Institute of PhysicsDownloaded 24 Mar 2009 to 140.105.16.64. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp
013509-2 Drissi, Jehjouh, and Saidi J. Math. Phys. 50, 013509 2009iiSecond, we extend the generalized MacMahon functions S 3 q and T 3 q,t by implementingboundary conditions captured by the strict notice that refers to a 2d-partition and ˆ toa strict 2d-partition. The hat is sometimes dropped out for simplicity of notations 2dpartitions ˆ , ˆ , and ˆ. The resulting S ˆ ˆ ˆq and T ˆ ˆ ˆq,t are also needed to complete thefollowing table:C q = known Eq. 2.1refining→shifting ↓ ↓ shiftingS ˆ ˆ ˆq = ... ?refining→R q,t = known: Eq. 2.6T ˆ ˆ ˆq,t = ... ?. 1.2The organization of this paper is as follows: In Sec. II, we give generalities on topologicalvertices. In particular, we review briefly the expression of the constructions of the standardtopological vertex C q: the refined one R q,t and the shifted MacMahon functionS 3 q. In Sec. III, we derive the explicit expression of the shifted topological vertex S ˆ ˆ ˆqof Eq. 1.2. In Sec. IV, we compute the its refined version T ˆ ˆ ˆq,t. In Sec. V we give aconclusion, and in Sec. VI we collect some useful tools as an Appendix.II. TOPOLOGICAL VERTEX: A REVIEWIn this section, we review briefly some basic tools; in particular, the explicit expressions of thethree following topological vertices:1 The standard topological vertex denoted as C q.23The refined topological vertex R q,t. This is a two parameter generalization of C q.The standard 3d-MacMahon function C 3 q, its refined version R 3 q,t as well as the shifted3d-MacMahon function S 3 q obtained in Ref. 15.These objects have interpretations in a the topological string A-model in which q=e −g s withg s being the topological string coupling constant. b The statistical mechanical models in whichthe parameter q describes the Boltzmann weight q=e −1/KT with T being absolute temperature. 14,19A. Vertex C „q…Following Ref. 3, the standard topological vertex C q, with boundary conditions in thex i ,x j orthogonal planes of Z 3 lattice given by the 2d partitions ,,, can be defined in thetransfer matrix method as follows:C q = f t q L 0 + q − q 0 Lt=0−1t=− − q −t q L 0,where is a generic 2d partition state a Young diagram, t its transpose and L 0 = with being the number of boxes of the Young diagram. The operators x are vertex operators ofthe c=1 bosonic CFT 2 , whose explicit expressions can be found in Ref. 14, and f is a realnumber given byf = q1/22 + 2 + 2 q n+nn=01−q n n.2.12.2For the particular case where there is no boundary condition, i.e., ===, the vertex C qcoincides exactly with the 3d-MacMahon function C 3 ,Downloaded 24 Mar 2009 to 140.105.16.64. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp
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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 topologiquesoù F es
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3.3 Invariants topologiquesDans le
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3.3 Invariants topologiquesL’acti
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3.4 Modèle B et espace twistorielI
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H. Jehjouhprojectif complexe -PT d
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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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L.B. Drissi et al. / Nuclear Physic
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(Γ + (z) = exp −i ∑ )1n z−n
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(∏ ∞(Υ − (q) = Ω − qk ))k
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5.1 Formalisme du Vertex topologiqu
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5.3 Vertex Topologique et Théorie
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H. JehjouhL’amplitude des produit
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176H. Jehjouh
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Page 277 and 278: Bibliographie[1] J. Polchinski, Str
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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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