Abstract
We study the local Gromov-Witten invariants of \( \mathcal{O} \)(k)⊕\( \mathcal{O} \)(−k−2) → ℙ1 by localization techniques and the Marino-Vafa formula, using suitable circle actions. They are identified with the equivariant Riemann-Roch indices of some power of the determinant of the tautological sheaves on the Hilbert schemes of points on the affine plane. We also compute the corresponding Gopakumar-Vafa invariants and make some conjectures about them.
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