Quasimap Wall-crossings and Mirror Symmetry

TL;DR

This paper establishes a wall-crossing formula for epsilon-stable quasimaps to GIT quotients, proving it for complete intersections in projective space, and relates it to Gromov-Witten invariants and mirror symmetry, especially for the quintic threefold.

Contribution

It introduces a new wall-crossing formula for quasimap invariants and proves it in general for complete intersections, connecting to mirror symmetry and BCOV theory.

Findings

Proved wall-crossing formula for epsilon-stable quasimaps

Established relation between Gromov-Witten and quasimap potentials

Provided geometric interpretation of BCOV B-model partition function

Abstract

We state a wall-crossing formula for the virtual classes of epsilon-stable quasimaps to GIT quotients and prove it for complete intersections in projective space, with no positivity restrictions on their first Chern class. As a consequence, the wall-crossing formula relating the genus g descendant Gromov-Witten potential and the genus g epsilon-quasimap descendant potential is established. For the quintic threefold, our results may be interpreted as giving a rigorous and geometric interpretation of the holomorphic limit of the BCOV B-model partition function of the mirror family.