Crepant Transformation Correspondence For Toric Stack Bundles

TL;DR

This paper develops a genus-zero crepant transformation framework for toric stack bundles connected by crepant wall-crossings on their toric fibers. It constructs a degree-preserving symplectic transformation $\mathbb{U}$ in Givental’s formalism that aligns the $I$-functions after Mellin–Barnes analytic continuation, and connects this with a Fourier-Mukai transform on equivariant $K$-theory. The approach parallels and extends Coates–Iritani–Jiang’s crepant transformation philosophy to the setting of toric stack bundles, using toric stack bundle mirror data and localization to perform the required analytic continuations. The main result yields a commutative diagram tying genus-zero Gromov–Witten theory of the bundles to their Fourier–Mukai-equivalent model, offering new insight into the interplay between GW theory, mirror symmetry, and derived-category transforms for toric stacks.

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.

Figures (1)

• Figure 1: text of caption

Theorems & Definitions (24)

• Theorem 1
• Remark 2.1
• Theorem 3.1
• Lemma 3.2
• proof
• Remark 3.3
• proof : Proof of Theorem \ref{['THM']}
• Lemma 3.4
• proof
• Theorem 3.5