Two point extremal Gromov-Witten invariants of Hilbert schemes of points on surfaces

Given an algebraic surface $X$, the Hilbert scheme $X^{[n]}$ of $n$-points on $X$ admits a contraction morphism to the $n$-fold symmetric product $X^{(n)}$ with the extremal ray generated by a class $\beta_n$ of a rational curve. We determine the two point extremal GW-invariants of $X^{[n]}$ with respect to the class $d\beta_n$ for a simply-connected projective surface $X$ and the quantum first Chern class operator of the tautological bundle on $X^{[n]}$. The methods used are vertex algebraic description of $H^*(X^{[n]})$, the localization technique applied to $X=\mathbb P^2$, and a generalization of the reduction theorem of Kiem-J. Li to the case of meromorphic 2-forms.