Gromov–Witten theory of $\mathcal{A}_{n}$–resolutions

Abstract

We give a complete solution for the reduced Gromov–Witten theory of resolved surface singularities of type $A_n$, for any genus, with arbitrary descendent insertions. We also present a partial evaluation of the $T$–equivariant relative Gromov–Witten theory of the threefold ${\mathcal{A}}_n\times P^1$ which, under a nondegeneracy hypothesis, yields a complete solution for the theory. The results given here allow comparison of this theory with the quantum cohomology of the Hilbert scheme of points on the ${\mathcal{A}}_n$ surfaces. We discuss generalizations to linear Hodge insertions and to surface resolutions of type $D,E$. As a corollary, we present a new derivation of the stationary Gromov–Witten theory of $P^1$.