Gromov-Witten invariants of $\mathrm{Sym}^d\mathbb{P}^r$

TL;DR

This paper introduces a graph-sum algorithm to compute genus-$g$ Gromov-Witten invariants of symmetric product orbifolds, linking them to Hurwitz-Hodge integrals and establishing a partial mirror theorem in genus zero.

Contribution

It presents a novel algorithm expressing Gromov-Witten invariants via Hurwitz-Hodge integrals and proves a partial mirror theorem for symmetric products of projective space.

Findings

Algorithm expresses invariants in terms of Hurwitz-Hodge integrals

Partial mirror theorem established in genus zero

Conditional proof of a conjectural power series equality

Abstract

We give a graph-sum algorithm that expresses any genus-$g$ Gromov-Witten invariant of the symmetric product orbifold $\mathrm{Sym}^d\mathbb{P}^r:=[(\mathbb{P}^r)^d/S_d]$ in terms of "Hurwitz-Hodge integrals" -- integrals over (compactified) Hurwitz spaces. We apply the algorithm to prove a partial mirror theorem for $\mathrm{Sym}^d\mathbb{P}^r$ in genus zero. The theorem states that a generating function of Gromov-Witten invariants of $\mathrm{Sym}^d\mathbb{P}^r$ is equal to an explicit power series $I_{\mathrm{Sym}^d\mathbb{P}^r},$ conditional upon a conjectural combinatorial identity. This is a first step towards proving Ruan's Crepant Resolution Conjecture for the resolution $\mathrm{Hilb}^{(d)}(\mathbb{P}^2)$ of the coarse moduli space of $\mathrm{Sym}^d\mathbb{P}^2.$