Mirror Symmetry for Stable Quotients Invariants

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.