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

013509-9 Refining the shifted topological vertex J. Math. Phys. 50, 013509 2009T diag = t + t i q i− − q j−1 t − j ti0j0Then using q L 0=2 l q and Eq. 3.17, we end with.4.14withdiag = Z T t − q − t − + t −t q − ,j0i04.15 q,t =2 −l−l q −/2 t −/2 , q,t = q t /2 .4.16Using Eq. 3.20 and the skew Schur functions P t / and Q / , the partition function 4.11 readsaswhereT =h˜ Z stricttq /2P t /q − t − Q / t −t q − ,h˜ q,t = q,t g t,q,=2 −l−l q −nt −/2 t −n−/2 .4.174.18To determine the factor Z, we need two data: first we use the identity T =Z q,t andsecond, we require that Z = =T 3 q,t, asinEq.4.1. We findZ q,t = q t 2 /22 −l P tt − 1+qj−1 t i1−q j−1 t i.4.19j,i=1This ends the proof of Eq. 4.2.Notice that K q,t can be also put in the closed formK = q t 2 + 2 /2t k/2 P tt − tq +−/2P t /x , Q / y , ,4.20with x , =q − t − , y , =t −t q − and the property T =T K as well as the normalizationK =1.V. CONCLUSIONIn this paper we have studied the refining and the shifting properties of the standard topologicalvertex C . After having reviewed some basic properties on the following:1 The standard vertex C q and its refined version R used in the framework of topologicalstrings.2 The shifted MacMahon function S 3 q used in BKP hierarchy,we have completed the missing relations in Eqs. 1.1 and 1.2. In particular, we havederived the explicit expressions of the following:aThe shifted topological vertex S ˆ ˆ ˆq with boundary conditions given by generic strict 2dpartitions. The shifted MacMahon function S 3 q, given by Eq. 2.9 and first obtained inRef. 15, follows by putting ˆ =ˆ =ˆ =.Downloaded 24 Mar 2009 to 140.105.16.64. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp