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Local Gromov-Witten invariants and tautological sheaves on Hilbert schemes. (English) Zbl 1215.14056

Summary: We study the local Gromov-Witten invariants of \(\mathcal{O} (k)\oplus \mathcal{O}( - k - 2) \rightarrow \mathbb P^{1}\) by localization techniques and the Mariño-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.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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