Gromov-Witten theory of A_n-resolutions

TL;DR

This paper provides a comprehensive solution for the reduced Gromov-Witten theory of A_n surface resolutions, enabling comparisons with quantum cohomology and extending to related theories and surface types.

Contribution

It offers a complete solution for the reduced Gromov-Witten theory of A_n resolutions for all genera and descendent insertions, and explores connections to other geometric theories.

Findings

Complete solution for A_n surface Gromov-Witten theory.

Partial evaluation of T-equivariant relative Gromov-Witten theory of A_n x P^1.

New derivation of stationary Gromov-Witten theory of P^1.

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 A_n x 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 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.