Double and Triple Givental's J-functions for Stable Quotients Invariants

TL;DR

This paper extends mirror formulas to stable quotients invariants, revealing their unique properties and connections to BPS counts and twisted Hurwitz numbers, thus broadening the understanding of mirror symmetry in algebraic geometry.

Contribution

It develops mirror formulas for double and triple Givental's J-functions in the stable quotients setting, highlighting their distinct properties from Gromov-Witten invariants.

Findings

Stable quotients invariants do not satisfy divisor, string, or dilaton relations.

They exhibit integrality properties similar to genus 0 Gromov-Witten invariants of Calabi-Yau manifolds.

Mirror formulas relate stable quotients invariants to BPS counts and twisted Hurwitz numbers.

Abstract

We use mirror formulas for the stable quotients analogue of Givental's J-function for twisted projective invariants obtained in a previous paper to obtain mirror formulas for the analogues of the double and triple Givental's J-functions (with descendants at all marked points) in this setting. We then observe that the genus 0 stable quotients invariants need not satisfy the divisor, string, or dilaton relations of the Gromov-Witten theory, but they do possess the integrality properties of the genus 0 three-point Gromov-Witten invariants of Calabi-Yau manifolds. We also relate the stable quotients invariants to the BPS counts arising in Gromov-Witten theory and obtain mirror formulas for certain twisted Hurwitz numbers.