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

013509-3 Refining the shifted topological vertex J. Math. Phys. 50, 013509 2009C 3 q = q L 0 + 1q 0 Lt=0−1t=−The function C 3 , which reads explicitly as, see also Eq. 1.1,C 3 q = n=11 nn1−q − 1q L 0 .2.32.4has several interpretations. It is the amplitude of the A-model topological closed string on C 3 ; i.e.,C 3 =C . It is also the generating function of 3d partitions,C 3 q =3d-partitions Likewise, the vertex C q inherits the interpretations of C 3 q with a slight generalization andmore power since it allows gluing 1 to topological amplitudes of all non compact toric CY3s.It is the partition function of A-model topological string of C 3 with open strings on boundariesand, up on using the gluing method, 1 it allows to compute the partition function of toric CY3s.C q has also a combinatorial interpretation in terms of the generating function of the planepartitions with the boundary conditions ,, where , , and are 2d-partitions.The explicit expression of C can be exhibited in different, but equivalent forms. Its expressionin terms of the product of three Schur functions can be found in Refs. 3 and 8.q .2.51. ExtensionsTwo kinds of generalizations of the topological vertex C have been considered in literature.These are as follows:i The refined vertex R q,t having a connection with Nekrasov’s 11 partition function ofSUN gauge theories and with the link invariants. 13ii The shifted MacMahon function S 3 q used in BKP hierarchy. 15 The S q extension ofthe shifted vertex by implementing boundary conditions was not computed before; it is aresult of the present paper.Let us give some details on R and S 3 ; then we turn back to the computation of S q.B. Refined vertex: R „q,t…The refined topological vertex R q,t is a two parameter extension of C q. As notedbefore, it has a topological string interpretation in connection with Nekrasov’s instantons. Itsexplicit expression has been first derived by Iqbal et al. and can be expressed in different butequivalent ways. It is given, in the transfer matrix method, byR q,t =r t t L 0 + q − it 0 Lt=0where , q L 0, and x are as before, and−1t=− − q − j t q L 0,2.6r = q1/22 + 2 + 2 1−q k−1 t l .2.7q nt t n k,l=1For the particular case ===, the vertex R q coincides exactly with the refined3d-MacMahon function mentioned in Sec. I 1.1,Downloaded 24 Mar 2009 to 140.105.16.64. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp