Super Macdonald polynomials and BPS state counting on the blow-up

TL;DR

This paper connects super Macdonald polynomials with BPS state counting on the blow-up of the projective plane, using localization techniques and quantum algebra actions to reveal new mathematical structures.

Contribution

It introduces a novel framework linking super Macdonald polynomials to BPS state counting via equivariant localization and quantum algebra representations.

Findings

Defined Nekrasov factors for super partitions

Computed matrix elements of quantum toroidal algebra actions

Confirmed consistency with Pieri rule of super Macdonald polynomials

Abstract

We explore the relation of the super Macdonald polynomials and the BPS state counting on the blow-up of $\mathbb{P}^2$, which is mathematically described by framed stable perverse coherent sheaves. Fixed points of the torus action on the moduli space of BPS states are labeled by super partitions. From the equivariant character of the tangent space at the fixed points we can define the Nekrasov factor for a pair of super partitions, which is used for the localization computation of the partition function. The Nekrasov factor also allows us to compute matrix elements of the action of the quantum toroidal algebra of type $\mathfrak{gl}_{1|1}$ on the $K$ group of the moduli space. We confirm that these matrix elements are consistent with the Pieri rule of the super Macdonald polynomials.