Instanton counting, Macdonald function and the moduli space of D-branes

TL;DR

Awata and Kanno link Nekrasov's partition function in the $\Omega$-background to the D-brane moduli space via geometric engineering and Gopakumar-Vafa invariants, showing the instanton expansion for $SU(2)$ factorizes into $SU(2)_L \times SU(2)_R$ characters. They demonstrate consistency of the spin content with Lefschetz actions on the moduli space of D2-branes on local ${\bf F}_0$ up to two instantons, and propose a refined two-parameter topological vertex built from Macdonald functions to generate equivariant $\chi_y$ and elliptic genera of Hilb^n ${\mathbb C}^2$. The work develops a Macdonald-based refinement of the topological vertex, enabling exact diagrammatic computation of refined BPS counts in five and six dimensions and establishing key Macdonald identities in the accompanying appendices. Together, these results provide a systematic framework for refined BPS state counting and connect gauge theory instantons, topological strings, and Hilbert schemes of points.

Abstract

We argue the connection of Nekrasov's partition function in the Ωbackground and the moduli space of D-branes, suggested by the idea of geometric engineering and Gopakumar-Vafa invariants. In the instanton expansion of N=2 SU(2) Yang-Mills theory the Nakrasov's partition function with equivariant parameters ε_1, ε_2 of toric action on C^2 factorizes correctly as the character of SU(2)_L \times SU(2)_R spin representation. We show that up to two instantons the spin contents are consistent with the Lefschetz action on the moduli space of D2-branes on (local) F_0. We also present an attempt at constructing a refined topological vertex in terms of the Macdonald function. The refined topological vertex with two parameters of T^2 action allows us to obtain the generating functions of equivariant χ_y and elliptic genera of the Hilbert scheme of n points on C^2 by the method of topological vertex.