On the polynomiality of orbifold Gromov--Witten theory of root stacks

Abstract

In [TY18], higher genus Gromov-Witten invariants of the stack of r-th roots of a smooth projective variety X along a smooth divisor D are shown to be polynomials in r. In this paper we study the degrees and coefficients of these polynomials.

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

“…It gives another explanation for the degree bound of orbifold invariants of X D,r in [TY20]. In [TY20], it is proved that genus (g + 1) orbifold invariants of root stacks (without mid-ages) are polynomials of degree bounded by 2g + 1. By loop axiom for orbifold Gromov-Witten theory, they can be written as sum of genus g orbifold invariants.…”

“…Let d 0 be the positive integer in Lemma 2.1. When k a ≤ d 0 , by [TY18], [FWY19] and [TY20], there is a sufficiently large r such that the cycle…”

“…By [FWY19] and [TY20], for sufficiently large r, invariants with k a ≤ d 0 will be polynomials in r of degree 2g −1 and the constant terms are relative invariants with negative contact orders. As k a gets bigger, the proof of [FWY19] and [TY20] will not work eventually, because we will need bigger r. They are polynomials in r of degree 2g instead of 2g − 1. It gives another explanation for the degree bound of orbifold invariants of X D,r in [TY20].…”

“…As k a gets bigger, the proof of [FWY19] and [TY20] will not work eventually, because we will need bigger r. They are polynomials in r of degree 2g instead of 2g − 1. It gives another explanation for the degree bound of orbifold invariants of X D,r in [TY20]. In [TY20], it is proved that genus (g + 1) orbifold invariants of root stacks (without mid-ages) are polynomials of degree bounded by 2g + 1.…”

“…Results in [ACW17], [TY18] and [TY20] stated that, for sufficiently large r, genus g orbifold Gromov-Witten invariants with small ages are polynomials in r with degree bounded by max{2g − 1, 0} and the constant terms are the corresponding relative Gromov-Witten invariants with positive contact orders. On the other hand, by [FWY20], [FWY19] and [TY20], genus g orbifold Gromov-Witten invariants with large ages, after multiplying by suitable power of r, are polynomials in r with degree bounded by max{2g − 1, 0} and the constant terms are the corresponding relative Gromov-Witten invariants with negative contact orders defined in [FWY20] and [FWY19]. In particular, genus zero orbifold invariants are equal to genus zero relative invariants when r is sufficiently large by [ACW17], [TY18] and [FWY20].…”