# Orbifold quantum cohomology of the symmetric product of A_r

## TL;DR

This paper computes the quantum cohomology ring of the symmetric product of A_r surfaces, confirming the Crepant Resolution Conjecture and linking multiple Gromov-Witten theories under certain assumptions.

## Contribution

It provides explicit calculations of quantum multiplication operators and establishes equivalences among various Gromov-Witten theories for symmetric products of A_r.

## Key findings

- Quantum multiplication operators determined by divisor classes.
- Confirmation of the Crepant Resolution Conjecture for [Sym^n(A_r)] and Hilb^n(A_r).
- Establishment of a tetrahedral relationship among Gromov-Witten theories.

## Abstract

Let A_r be the minimal resolution of the cyclic quotient singularity C^2/Z_{r+1}. We study the equivariant quantum cohomology ring of the n-fold symmetric product stack [Sym^n(A_r)] of A_r. We calculate the operators of quantum multiplication by divisor classes. Under the assumption of the nonderogatory conjecture, these operators completely determine the ring structure, which provides an affirmative answer to the Crepant Resolution Conjecture on [Sym^n(A_r)] and Hilb^n(A_r). More strikingly, this allows us to complete a tetrahedron of equivalences relating the Gromov-Witten theories of [Sym^n(A_r)]/Hilb^n(A_r) and the relative Gromov-Witten/Donaldson-Thomas theories of A_r x P^1.