Higher genus relative and orbifold Gromov-Witten invariants of curves

Abstract

Given a smooth target curve $X$, we explore the relationship between Gromov-Witten invariants of $X$ relative to a smooth divisor and orbifold Gromov-Witten invariants of the $r$-th root stack along the divisor. We proved that relative invariants are equal to the $r^0$-coefficient of the corresponding orbifold Gromov-Witten invariants of $r$-th root stack for $r$ sufficiently large. Our result provides a precise relation between relative and orbifold invariants of target curves generalizing the result of Abramovich-Cadman-Wise to higher genus invariants of curves. Moreover, when $r$ is sufficiently large, we proved that relative stationary invariants of $X$ are equal to the orbifold stationary invariants in all genera. Our results lead to some interesting applications: a new proof of genus zero equality between relative and orbifold invariants of $X$ via localization; a new proof of the formula of Johnson-Pandharipande-Tseng for double Hurwitz numbers; a version of GW/H correspondence for stationary orbifold invariants.

Results of this article has been merged into arXiv:1806.11082

References in corpus (2)

• Equivariant Gromov-Witten theory of one dimensional stacks
• Double ramification cycles on the moduli spaces of admissible covers

Cited by in corpus (6)

• Double ramification cycles with target varieties
• Structures in genus-zero relative Gromov--Witten theory
• Higher genus relative Gromov--Witten theory and DR-cycles
• Pixton's formula and Abel-Jacobi theory on the Picard stack
• Relative Gromov-Witten invariants and the enumerative meaning of mirror maps for toric Calabi-Yau orbifolds
• Mirror theorems for root stacks and relative pairs