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

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