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

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 ${\mathrm{Sym}}^d{\mathbb{P}}^2.$

42 pages, comments welcome. Substantial corrections and clarifications to the proof of our main application, Theorem 6.3, which has also been slightly weakened. Minor errors corrected elsewhere. Version for publication

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