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

013509-5 Refining the shifted topological vertex J. Math. Phys. 50, 013509 2009withS = P t /q −− Q / q −t− Z h ,strict 2d = 1 , ..., i , ... ,,3.5 k = 1 2 − k,3.6as well asandn = 1 2 t 2 − ,h =2 −l−l q −nt −n q −/2−/2 ,q −− = q − 1 − 1, ...,q − i − i,...,3.7Z = 2−l P tq − q +/2+nt n=11+q n1−q nn .The function P x is the Schur function associated with the strict 2d partition ; it is defined as3.8 − x =strict 2d partition P / x,where / is the complement of in . We also have the orthogonality relation3.9Q x =2 −l P x,Q x,P x = ,3.10where = i i i.To establish this result, we consider shifted 3d-partitions 3 inside of a cube with sizeN 1 N 2 N 3 and boundary conditions given by the strict 2d-partitions ,,. More precisely, thestrict 2d-partition belongs to the plane x 2 ,x 3 of the ambient real three-dimensional space, belongs to the plane x 3 ,x 1 and to the plane x 1 ,x 2 ,Then proceed by steps as follows:Step 1: Compute the perpendicular partition function S q by using the transfer matrixapproach. 3 This method has been used for calculating the topological vertex C of the A-modeltopological string on C 3 which lead to Eqs. 2.1–2.4. S reads in terms of products x asfollows:S = q−n−nt q N 2 +N 1 t N 1 N 2q L 0 + q − jq 0L − q − i t q L 0.j=1i=13.11By using the relation q −kL 0 zq kL 0= zq k and q L 0=2 l q , the function S can bebrought towithS = t j=1N 1 + q − j − jN 2 − q − i t − i ,i=13.12Downloaded 24 Mar 2009 to 140.105.16.64. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp