In this paper, we define genus‐zero relative Gromov–Witten invariants with negative contact orders. Using this, we construct relative quantum cohomology rings and Givental formalism. A version of Virasoro constraints also follows from it.

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Given a smooth projective variety X with a simple normal crossing divisor D := D 1 + D 2 + ... + D n , where D i ⊂ X are smooth, irreducible and nef. We prove a mirror theorem for multi-root stacks X D, r by constructing an I-function, a slice of Givental's Lagrangian cone for Gromov-Witten theory of multi-root stacks. We provide three applications: (1) We show that some genus zero invariants of X D, r stabilize for sufficiently large r. (2) We state a generalized local-log-orbifold principle conjecture and prove a version of it. (3) We show that regularized quantum periods of Fano varieties coincide with classical periods of the mirror Landau-Ginzburg potentials using orbifold invariants of X D, r .

We extend the definition of relative Gromov–Witten invariants with negative contact orders to all genera. Then we show that relative Gromov–Witten theory forms a partial CohFT. Some cycle relations on the moduli space of stable maps are also proved.

Abstract. We study Givental's Lagrangian cone for the quantum orbifold cohomology of toric stack bundles. Using Gromov-Witten invariants of the base and combinatorics of the toric stack fibers, we construct an explicit slice of the Lagrangian cone defined by the genus 0 Gromov-Witten theory of a toric stack bundle.

Given a smooth projective variety X with a smooth nef divisor D and a positive integer r, we construct an I-function, an explicit slice of Givental's Lagrangian cone, for Gromov-Witten theory of the root stack X D,r . As an application, we also obtain an I-function for relative Gromov-Witten theory following the relation between relative and orbifold Gromov-Witten invariants.

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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.