Local Gromov-Witten invariants and tautological sheaves on Hilbert schemes

We study the local Gromov-Witten invariants of O(k) circle plus O(-k -2) -> P-1 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.