Local Gromov-Witten invariants and tautological sheaves on Hilbert schemes

Abstract

We study the local Gromov-Witten invariants of O(k) ⊕ O(−k − 2) → P 1 by localization techniques and the Mariño-Vafa formula, using suitable circle actions. They are identified with the equivariant RiemannRoch 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.

Localized Standard Versus Reduced Formula and Genus 1 Local Gromov–Witten Invariants

“…where α (v,e) = α i,j if e = {v, v ′ } with m(v ′ ) = j, and ψ (v,e) is the ψ-class associated to the marked point on M 0,val(v) corresponding to the edge e. The explicit form of Cont Γ;e µ 1 , · · · , µ |J| X 1,J,d can be computed by the holomorphic Lefschetz formula ( [3]), see e.g., [31]; we will not spell out the general formula for this since we don't need it. For X of the form (8), the explicit form for Cont Γ;e µ 1 , · · · , µ |J| X 1,J,d is (134).…”

“…can be computed by the holomorphic Lefschetz formula ([3]), see e.g., [31]; we will not spell out the general formula for this since we don't need it. For X of the form (8), the explicit form for…”

K-Theory of Hilbert Schemes as a Formal Quantum Field Theory

“…When the rank is one, the relevant moduli spaces are the Hilbert schemes (C 2 ) [n] , the tautological bundle is the tautological bundle ξ n induced from the trivial bundle O C 2 . The following result was proved in Yang-Zhou [64] in 2011:…”

“…(For other applications of Hilbert schemes to curve counting problem, see [27] and the references therein.) The motivation behind [32] and [64] is the computation of Gopakumar-Vafa invariants of the corresponding toric Calabi-Yau 3-folds. The idea is to find analogues of Göttsche's formula for K-theoretical intersection numbers on Hilbert schemes.…”

Crepant Resolutions, Quivers and GW/NCDT Duality

“…By a result due to Laufer [23], when the exceptional set is P 1 , its normal bundle is isomorphic to one of the following three bundles: O(−1) ⊕ O(−1), O(−2) ⊕ O, and O(−3) ⊕ O (1). The first case can be realized by the well-known resolved conifold, and the their local Gromov-Witten invariants are well-known [17,12]; the other two case are realized by Laufer's examples [23], for their local Gromov-Witten invariants, and more generally, that of O(k) ⊕ O(−k − 2) → P 1 for k ≥ −1 see [36].…”