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

Robert Silversmith

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

Algebraic Geometry 2023-03-14 v3

Authors: Robert Silversmith

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

Keywords

Gromov-Witten invariants symmetric product orbifold Hurwitz-Hodge integrals graph-sum algorithm mirror theorem crepant resolution conjecture moduli space

Cite

@article{arxiv.1611.05941,
  title  = {Gromov-Witten invariants of $\mathrm{Sym}^d\mathbb{P}^r$},
  author = {Robert Silversmith},
  journal= {arXiv preprint arXiv:1611.05941},
  year   = {2023}
}

Comments

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

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

Abstract

Keywords

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