Journal
INTERNATIONAL JOURNAL OF MODERN PHYSICS A
Volume 24, Issue 12, Pages 2253-2306Publisher
WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S0217751X09043006
Keywords
BPS state; topological string; symmetric functions
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It has been argued that 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 previously proposed (arXiv:hep-th/0502061) is a building block of Nekrasov's partition function with two equivariant parameters. Compared with another refined topological vertex by Iqbal, Kozcaz and Vafa (arXiv: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.
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