Gromov-Witten invariants of 𝑆𝑦𝑚^{𝑑}ℙ^{𝕣}

Abstract

We give a graph-sum algorithm that expresses any genus- g g Gromov-Witten invariant of the symmetric product orbifold Sym d ⁡ P r ≔ \operatorname {Sym}^d\mathbb {P}^r≔ [ ( P r ) d / S d ] [(\mathbb {P}^r)^d/S_d] in terms of “Hurwitz-Hodge integrals” – integrals over (compactified) Hurwitz spaces. We apply the algorithm to prove a mirror-type theorem for Sym d ⁡ P r \operatorname {Sym}^d\mathbb {P}^r in genus zero. The theorem states that a generating function of Gromov-Witten invariants of Sym d ⁡ P r \operatorname {Sym}^d\mathbb {P}^r is equal to an explicit power series I Sym d ⁡ P r I_{\operatorname {Sym}^d\mathbb {P}^r} , conditional upon a conjectural combinatorial identity. This is a first step in the direction of proving Ruan’s Crepant Resolution Conjecture for the resolution H i l b ( d ) ( P 2 ) Hilb^{(d)}(\mathbb {P}^2) of the coarse moduli space of Sym d ⁡ P 2 . \operatorname {Sym}^d\mathbb {P}^2.