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

013509-11 Refining the shifted topological vertex J. Math. Phys. 50, 013509 2009FIG. 2. A strict plane partition.For illustration, see the example Fig. 2... −1 0 1 ... A3 −2 = 3, −1 = 4,3, 0 = 5,3, 1 = 3,2, 2 = 2, 3 = 1.2. Property of Schur function for strict partitionThe shifted topological vertex is defined by using skew Schur P and Q functions. 20–22 Theseare symmetric functions that appear in topological amplitudes and are defined by a sequence ofpolynomials P x 1 ,x 2 ,...,x n , nN, with the property= P / x 1 , ...,x n Tx T , 0, otherwise, A4where the sum is over all shifted Young tableaux of shape /. The skew Schur function Q / isrelated to P / as in Eqs. 3.9 and 3.10. We also havestrict Q xP y = i,j1+x iy j1−x i y j.The relation between the Schur function P for strict partition that we have used here above andthe usual Schur functions S ˜ for the double partition ˜ is given byA5S ˜t =2 −l P 2t 2,A6where t/2 is t 1 /2,t 3 /2,t 5 /2,... and P t/2= P t 1 /2,t 3 /2,t 5 /2,.... Notice that the doublepartition ˜ in Frobenuis notation reads in terms of the strict partition =n 1 ,n 2 ,...,n k as˜ = n1 ,n 2 , ...,n k n 1 −1,n 2 −1, ...,n k−1 −1.A71 M. Aganagic, A. Klemm, M. Marino, and C. Vafa, Commun. Math. Phys. 254, 425 20052 A. Iqbal and A.-K. Kashani-Poor, Adv. Theor. Math. Phys. 7, 457 2004.3 A. Okounkov, N. Reshetikhin, and C. Vafa, Prog. Math., 244, 597 2006.4 L. B. Drissi, H. Jehjouh, and E. H. Saidi, Nucl. Phys. B to be published.5 T. Graber and E. Zaslow, e-print arXiv:hep-th/0109075.6 M. Aganagic, A. Klemm, and C. Vafa, Z. Naturforsch., A: Phys. Sci. 57, 12002.7 D. Karp, C. Liu, and M. Marino, Geom. Topol. 10, 1152006.8 A. Iqbal, C. Kozcaz, and C. Vafa, e-print arXiv:hep-th/0701156.Downloaded 24 Mar 2009 to 140.105.16.64. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp