A mirror theorem for Gromov-Witten theory without convexity
Jun Wang
TL;DR
The paper advances genus-zero Gromov–Witten theory for complete intersections in proper toric Deligne–Mumford stacks by proving a mirror theorem that does not require convexity. It constructs a big I-function from quasimap theory and proves it sits on Givental's Lagrangian cone after a mirror transform, i.e., -z I(q,t,-z) lies on the cone and I = J under μ. The key mechanism is a pair of master spaces and virtual localization, which produce recursive relations (through auxiliary cycles) that implement a genus-zero quasimap wall-crossing, enabling a rigorous identification of I and J in non-convex settings. The method yields explicit I-functions for complete intersections in toric DM stacks and generalizes quantum Lefschetz-type results to non-convex line bundles, with an illustrative Corti-type example demonstrating the non-convex case.
Abstract
We prove a genus zero Givental-style mirror theorem for all complete intersections in proper toric Deligne-Mumford stacks, which provides an explicit slice called big $I-$function on Givental's Lagrangian cone for such targets. In particular, we remove a technical assumption called convexity needed in previous quantum Lefschetz theorem.