Strengthening the Cohomological Crepant Resolution Conjecture for Hilbert-Chow morphisms

TL;DR

This paper proves a strengthened version of Ruan's Cohomological Crepant Resolution Conjecture for Hilbert schemes of points on smooth toric surfaces, relating their Gromov-Witten invariants to those of symmetric product stacks.

Contribution

It establishes a SYM-HILB correspondence that relates Gromov-Witten invariants of symmetric products and Hilbert schemes without parameter specialization, strengthening existing conjectures.

Findings

Proves a SYM-HILB correspondence for smooth toric surfaces.

Strengthens Ruan's Cohomological Crepant Resolution Conjecture.

Provides a method to reconstruct cup products from orbifold invariants.

Abstract

Given any smooth toric surface S, we prove a SYM-HILB correspondence which relates the 3-point, degree zero, extended Gromov-Witten invariants of the n-fold symmetric product stack [Sym^n(S)] of S to the 3-point extremal Gromov-Witten invariants of the Hilbert scheme Hilb^n(S) of n points on S. As we do not specialize the values of the quantum parameters involved, this result proves a strengthening of Ruan's Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphism from Hilb^n(S) to Sym^n(S) and yields a method of reconstructing the cup product for Hilb^n(S) from the orbifold invariants of [Sym^n(S)].