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

013509-8 Drissi, Jehjouh, and Saidi J. Math. Phys. 50, 013509 2009We also haveh˜ q,t =2 −l−l q −nt −/2 t −n−/2 .Z t,q = q t 2 /22 −l P tt − i,j=1 1+qj−1 t i1−q j−1 t i.as well as q,t=q/t /2 . Notice that for q=t=1, q,t=1.To establish this result, we use the following steps.Step 1: Compute the refined expression T 3 q,t of the shifted 3d MacMahon’s function S 3 qin terms of the two parameters q and t. To that purpose, we start from the defining relation ofT 3 q,t by using strict 2d-partitions,T 3 q,t =strict 2d partition 4.44.52 p q a=1 −at a=1a−1, 4.6where we have used the diagonal slicing of shifted 3d-partitions in terms of the strict 2d-onesa as shown below = a,aZa = i,i+a .iNotice that the slices with a0 are weighted by the factor q a while the slices with a0 areweighted by t a .Then, we use the transfer matrix method which allows to express T 3 q,t as the amplitudeT q,t; that is,T q,t = 0 t L 0 + 1t 0 La=0−a=−1 − 1q L 00.By using q −kL 0 zq kL 0= zq k , we can also put T in the formT q,t = 0 + t i − q j−1 0.i0j0Next commuting the − ’s to the left of the + ’s by help of relations 3.17, we obtain 1+qj−1 t i1−q j−1 t i, T 3 q,t = T q,t = j=1i=14.74.84.94.10which reduces to S 3 q Eq. 3.16 by setting t=q.Step 2: To get the expression of the perpendicular partition function T for arbitrary boundaryconditions, we mimic the approach in Ref. 1 and factorize T as follows:T q,t = g t,q T diag ,where T diag stands for the diagonal partition function and g t,q given by4.11g t,q = q −nt t −n4.12describing the change from diagonal slicing to perpendicular one. To compute T diag , we use thetransfer matrix method. We first haveT diag = t t L 0 + q i 0− tL − t − t jq 0 L .i0j0By using q −kL 0 zq kL 0= zq k , we can bring it to the form4.13Downloaded 24 Mar 2009 to 140.105.16.64. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp