Genus zero BPS invariants for local P¹

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We study the equivariant version of the genus zero BPS invariants of the total space of a rank 2 bundle on P¹ 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.