Gromov--Witten invariants of root stacks with mid-ages and the loop axiom

Abstract

We study orbifold Gromov-Witten invariants of the r-th root stack XD,r with a pair of mid-ages when r is sufficiently large. We prove that genus g invariants with a pair of mid-ages are polynomials in r of degree 2g. Moreover, genus zero invariants with a pair of mid-ages are independent of the choice of mid-ages. As an application, we state a modified version of the loop axiom for relative Gromov-Witten theory. Contents 10 3.1. Invariants without large ages 10 3.2. Invariants with large ages 12 4. Loop axiom for relative Gromov-Witten theory 13 4.1. CohFT 13 4.2. Genus one 16 4.3. Higher genus 18 References 18

A Gromov-Witten theory for simple normal-crossing pairs without log geometry

“…It would be interesting to find a replacement of the loop axiom. Some results along this direction has been proved in [35] by studying orbifold Gromov-Witten invariants of finite root stacks with mid-ages.…”