Summary: 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 [the authors, Instanton counting, Macdonald function and the moduli space of \(D\)-branes. J. High Energy Phys. 0505, 039, 26 p., (2005), arXiv:hep-th/0502061] is a building block of Nekrasov’s partition function with two equivariant parameters. Compared with another refined topological vertex by A. Iqbal, C. Kozcaz and C. Vafa [arXiv:hep-th/0701156, 70 p. (2009)], 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 \(\mathbb{C}^2\).
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.