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

013509-7 Refining the shifted topological vertex J. Math. Phys. 50, 013509 2009 x j =strict 2d L / x j ,N 1 x j = L / x 1 ,x 2 ,...,j=1strict 2d with x=x 1 ,x 2 ,... and where we have set L −/ = P / and L + / =Q / .Then, the partition function S becomesS q = Z q−/2+n−/2+nt 2 l+l P t /q −t− Q / q −− .strict 2d 3.203.21To determine the factor Z, we first use the identity S =Z, then the cyclic propertyS =S =S which implies in turns that S =S ; from which we learn the followingresult:Z = q−/2−nt 1+qn2 l 1−q nnP tq − . 3.22n=1Step 3: The shifted MacMahon function S 3 q can be recovered from the above analysis byusing the identity S 3 q=S =Z.This ends the proof of Eq. 3.2. Notice that L q can be also put in the formL q = q k/2 P tq − strict 2d P t /q −− Q / q −t − ,where k=2 2 − t 2 is the Casimir associated to strict 2d partition.3.23IV. REFINING THE SHIFTED VERTEXIn this section, we derive the explicit expression of the refining version T q,t of theshifted topological vertex S q. This is a two parameters q and t with boundary conditionsgiven by the strict 2d partitions , and .Notice that like for R of Eq. 2.6, the function the refined-shifted topological vertex T is noncyclic with respect to the permutations of the strict 2d partitions ,,; T T T . It obeys, however, the propertiesT q,q = S q, T q,t = T 3 q,t. 4.1Proposition 2: The explicit expression of the refined-shifted topological vertex T reads asfollows:K t,q =f T 1+qj−1 t i1−q j−1 t i,4.2where the factor f =f q is the same as in Eq. 3.4.T =T q,t is the refining version of S q of Eq. 3.5; it is the perpendicular partitionfunction generating the strict plane partitions. Its explicit expression reads in terms of the skewSchur functions P t / and Q / as follows:i,j=1T =h˜q,t Z t,qstrict 2d partitions q,tP t /q − t − Q / t −t q − ,4.3where h˜ is the refinement of h 3.7 and it is given byDownloaded 24 Mar 2009 to 140.105.16.64. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp