Refined BPS state counting from Nekrasov's formula and Macdonald functions

TL;DR

This paper introduces a new refined topological vertex based on Macdonald functions, providing a diagrammatic method to compute Nekrasov's partition function and exploring its relation to homological invariants of links.

Contribution

It presents a refined topological vertex using Macdonald functions, connecting Nekrasov's formula with geometric and algebraic structures in string theory.

Findings

The refined vertex is expressed via Macdonald symmetric functions.

Diagrammatic rules for partition function computation are established.

The vertex's transformation under flop operations links to Hopf link invariants.

Abstract

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.