Mirror symmetric Gamma conjecture for toric GIT quotients via Fourier transform

TL;DR

This paper proves the mirror symmetric Gamma conjecture for toric Fano orbifolds by computing Fourier transforms of quantum cohomology central charges, linking them to mirror Landau-Ginzburg models and deformation of cycles.

Contribution

It introduces a novel Fourier transform approach to connect quantum cohomology central charges with mirror Landau-Ginzburg models for toric Fano orbifolds.

Findings

Provides a new proof of the mirror symmetric Gamma conjecture for toric Fano orbifolds.

Establishes a link between Fourier transforms of quantum cohomology and mirror oscillatory integrals.

Shows deformation of integration cycles corresponds to parameter changes in the mirror symmetry context.

Abstract

Let $\mathcal X=[(\mathbb C^r\setminus Z)/G]$ be a toric Fano orbifold. We compute the Fourier transform of the $G$-equivariant quantum cohomology central charge of any $G$-equivariant line bundle on $\mathbb C^r$ with respect to certain choice of parameters. This gives the quantum cohomology central charge of the corresponding line bundle on $\mathcal X$, while in the oscillatory integral expression it becomes the oscillatory integral in the mirror Landau-Ginzburg mirror of $\mathcal X$. Moving these parameters to real numbers simultaneously deforms the integration cycle to the mirror Lagrangian cycle of that line bundle. This computation produces a new proof the mirror symmetric Gamma conjecture for $\mathcal X$.