Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2008-05-02
Int.J.Mod.Phys.A24:2253-2306,2009
Physics
High Energy Physics
High Energy Physics - Theory
56 pages, 13 figures; v2 a few improvements, typos fixed, a reference added; v3 Appendix A revised, typos corrected; v4 equati
Scientific paper
10.1142/S0217751X09043006
It has been argued that the Nekrasov's partition function gives the generating function of refined BPS state counting in the compactification of M theory on local Calabi-Yau spaces. We show that a refined version of the topological vertex we proposed before (hep-th/0502061) is a building block of the Nekrasov's partition function with two equivariant parameters. Compared with another refined topological vertex by Iqbal-Kozcaz-Vafa (hep-th/0701156), our refined vertex is expressed entirely in terms of the specialization of the Macdonald symmetric functions which is related to the equivariant character of the Hilbert scheme of points on C^2. We provide diagrammatic rules for computing the partition function from the web diagrams appearing in geometric engineering of Yang-Mills theory with eight supercharges. Our refined vertex has a simple transformation law under the flop operation of the diagram, which suggests that homological invariants of the Hopf link are related to the Macdonald functions.
Awata Hidetoshi
Kanno Hiroaki
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