Genus zero BPS invariants for local P^1

TL;DR

This paper computes equivariant genus zero BPS invariants for a rank 2 bundle over P^1 using residue integrals on moduli spaces, confirming predictions from local Gromov-Witten theory.

Contribution

It introduces a method to compute equivariant genus zero BPS invariants via residue integrals and verifies these results against theoretical predictions.

Findings

Computed low-degree invariants matching Gromov-Witten predictions

Established a residue integral approach for stable sheaves

Confirmed the consistency of invariants with theoretical models

Abstract

We study the equivariant version of the genus zero BPS invariants of the total space of a rank 2 bundle on P^1 whose determinant is O(-2). We define the equivariant genus zero BPS invariants by the residue integrals on the moduli space of stable sheaves of dimension one as proposed by Sheldon Katz. We compute these invariants for low degrees by counting the torus fixed stable sheaves. The results agree with the prediction in local Gromov-Witten theory.