Crepant Transformation Correspondence For Toric Stack Bundles

TL;DR

This paper establishes a crepant transformation correspondence in genus zero Gromov-Witten theory for toric stack bundles, using symplectic transformations and Mellin-Barnes integrals to relate different crepant models.

Contribution

It constructs a symplectic transformation that matches $I$-functions of toric stack bundles across crepant wall-crossings, linking Gromov-Witten invariants and $K$-theory via Fourier-Mukai.

Findings

Constructed a symplectic transformation for crepant wall-crossings.

Analytically continued $I$-functions using Mellin-Barnes integrals.

Linked Gromov-Witten theory with $K$-theory through Fourier-Mukai.

Abstract

We prove a crepant transformation correspondence in genus zero Gromov-Witten theory for toric stack bundles related by crepant wall-crossings of the toric fibers. Specifically, we construct a symplectic transformation that identifies $I$-functions toric stack bundles suitably analytically continued using Mellin-Barnes integral approach. We compare our symplectic transformation with a Fourier-Mukai isomorphism between the $K$-groups.