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.

No abstract

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.