Quantum Cohomology of Toric Blowups and Landau-Ginzburg Correspondences

TL;DR

This paper establishes a genus zero correspondence between the Gromov-Witten theory of certain orbifolds and their blowups, extending crepant transformation conjectures to non-crepant cases and applying these results to Landau-Ginzburg models.

Contribution

It generalizes the crepant transformation conjecture to non-crepant blowups using asymptotic expansion techniques and applies this to Landau-Ginzburg/Calabi-Yau correspondences.

Findings

Established a genus zero correspondence for orbifold blowups.

Extended crepant transformation conjecture to non-crepant settings.

Proved LG/Fano and LG/general type correspondences for hypersurfaces.

Abstract

We establish a genus zero correspondence between the equivariant Gromov-Witten theory of the Deligne-Mumford stack $[\mathbb{C}^N/G]$ and its blowup at the origin. The relationship generalizes the crepant transformation conjecture of Coates-Iritani-Tseng and Coates-Ruan to the discrepant (non-crepant) setting using asymptotic expansion. Using this result together with quantum Serre duality and the MLK correspondence we prove LG/Fano and LG/general type correspondences for hypersurfaces.