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

013509-4 Drissi, Jehjouh, and Saidi J. Math. Phys. 50, 013509 2009R 3 q,t = 1−q k−1 t l .k,l=12.8By setting t=q back in 2.8, we get the standard C 3 q relation. The explicit expression of therefined R q,t in terms of Schur functions can be found in Ref. 8.C. Shifted S 3 „q…The shifted 3d MacMahon S 3 q is the generating functional of strict plane partitions. Theexplicit expression of S 3 q has been derived by Foda and Wheeler by using transfer matrixmethod. It reads asS 3 q = n=1 n1+qn n, 2.91−qand has an interpretation in the BKP hierarchy of the so-called neutral free fermions. It wasclaimed in Ref. 15 that S 3 q could be relevant to the topological string dual to ON Chern–Simon theory in the limit N→.III. SHIFTED TOPOLOGICAL VERTEXThe expression Eq. 2.9 of the shifted topological vertex has been derived in the absence ofany kind of boundary conditions. Here, we want to complete this result by considering the derivationof the shifted topological vertex S with generic boundary conditions with the property,S = S q, S = S 3 q. 3.1Notice that S q generates the shifted 3d partitions with boundary conditions given by the strict2d partitions ,, along the axis x 1 ,x 2 ,x 3 . For the definitions of the shifted 3d and strict 2dpartitions, see the Appendix.The main result of this section is collected in the following proposition where some terminologyhas been borrowed from: 8Proposition 1: The perpendicular shifted topological vertex S q with generic boundaryconditions, given by three strict 2d-partitions ,,, reads as follows:L q = f S 1−qn1+q nn, 3.2which can be also put in the normalized formn=1In relation 3.2, the numerical factor f isS = S L ,L q =1.3.3f =2 l+l+l q 1/22 + 2 + 2 ,f q =1,where 2 = i i 2 and l is the length of the strict partition that is the number parts of the strict2d partition = 1 ,..., l ,0,....The function S q is the perpendicular partition function of shifted 3d-partitions. It reads interms of Schur functions P t / and Q / as follows:3.4Downloaded 24 Mar 2009 to 140.105.16.64. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp