Mirror theorems for root stacks and relative pairs

Relative Gromov-Witten invariants and the enumerative meaning of mirror maps for toric Calabi-Yau orbifolds

“…The third question is how to obtain a mirror theorem for relative pairs. In joint work with H. Fan and H.-H. Tseng [18], we proved a mirror theorem for root stacks X D,r . By taking suitable limits to the I-functions for X D,r , we obtained the I-functions for (X, D), hence a mirror theorem for relative pairs (X, D).…”

“…The computation of these invariants relies on the mirror theorem for relative pairs ( X , D ∞ ). Note that the mirror theorem for a pair (X, D) in [18] requires X to be a smooth projective variety and D to be a smooth nef divisor. We need a generalization of the result in [18] to orbifold pairs in order to compute relative invariants of the toric orbifolds ( X , D ∞ ).…”

“…Note that the mirror theorem for a pair (X, D) in [18] requires X to be a smooth projective variety and D to be a smooth nef divisor. We need a generalization of the result in [18] to orbifold pairs in order to compute relative invariants of the toric orbifolds ( X , D ∞ ). We briefly sketch the genus zero relative/orbifold correspondence with stacky targets in Appendix A.…”

“…We briefly sketch the genus zero relative/orbifold correspondence with stacky targets in Appendix A. In [18], the proof of the mirror theorem for pairs relies on orbifold quantum Lefschetz principle in [39] and the mirror theorem for toric stack bundles in [32]. Since orbifold quantum Lefschetz principle can fail for positive orbifold hypersurfaces [17] and the mirror theorem in [32] requires the toric stack bundles to have a smooth base, the result in Appendix A does not imply a generalization of the mirror theorem for general orbifold pairs.…”

“…However, as we are working on the toric Calabi-Yau orbifold X along with the toric divisor D ∞ , the root stack XD∞,r is still a toric orbifold. We can avoid the hypersurface construction in [18] and simply apply the mirror theorem for toric stacks in [16]. By taking a suitable limit as in [18,Section 4], we obtain a mirror theorem for the pair ( X , D ∞ ), see Theorem 3.11.…”