Mirror Symmetry for Stable Quotients Invariants

TL;DR

This paper demonstrates that stable quotients invariants for certain varieties are described by the same mirror hypergeometric series as Gromov-Witten invariants, revealing a deep connection between these theories and mirror symmetry.

Contribution

It establishes a mirror symmetry relation for stable quotients invariants, showing they match Gromov-Witten invariants under positivity conditions and linking them to hypergeometric series.

Findings

Stable quotients invariants are described by the same hypergeometric series as Gromov-Witten invariants in the Calabi-Yau case.

Certain twisted Hurwitz numbers are also governed by the same hypergeometric series.

The results confirm conjectures about the relationship between stable quotients and mirror symmetry.

Abstract

The moduli space of stable quotients introduced by Marian-Oprea-Pandharipande provides a natural compactification of the space of morphisms from nonsingular curves to a nonsingular projective variety and carries a natural virtual class. We show that the analogue of Givental's J-function for the resulting twisted projective invariants is described by the same mirror hypergeometric series as the corresponding Gromov-Witten invariants (which arise from the moduli space of stable maps), but without the mirror transform (in the Calabi-Yau case). This implies that the stable quotients and Gromov-Witten twisted invariants agree if there is enough "positivity", but not in all cases. As a corollary of the proof, we show that certain twisted Hurwitz numbers arising in the stable quotients theory are also described by a fundamental object associated with this hypergeometric series. We thus completely answer some of the questions posed by Marian-Oprea-Pandharipande concerning their invariants. Our results suggest a deep connection between the stable quotients invariants of complete intersections and the geometry of the mirror families. As in Gromov-Witten theory, computing Givental's J-function (essentially a generating function for genus 0 invariants with 1 marked point) is key to computing stable quotients invariants of higher genus and with more marked points; we exploit this in forthcoming papers.