Mirror symmetric Gamma conjecture for toric GIT quotients via Fourier transform

TL;DR

The work develops a Fourier-transform perspective on mirror symmetry for toric GIT quotients $\mathcal{X}$, connecting the $G$-equivariant quantum cohomology central charges on $\mathbb{C}^r$ to the non-equivariant central charges on $\mathcal{X}$ via an inverse Fourier transform of the B-model data. By analyzing the mirror Landau-Ginzburg model $W^{\mathcal{X}}$ and its equivariant perturbations, the authors show that the transformed central charges match the A-model expressions (involving the Gamma-class correction) and that this framework yields a rapid proof of the mirror symmetric Gamma conjecture for $\mathcal{X}$, without invoking GKZ-type PDEs. The construction relies on a detailed toric setup, including the extended nef and Mori cones, inertia/Chen-Ruan cohomology, and the CCC-based description of mirror cycles, culminating in a correspondence between central charges through a deformation to real Kähler parameters that encodes the SYZ mirror skeleton. This advances Teleman’s and Iritani’s program by providing a toric-quotient realization of Fourier-transform relations between equivariant and non-equivariant quantum data and clarifies the role of Euler and Gamma-class corrections in the integral-structure mirror map.

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 $\mathcal X$.

Figures (1)

• Figure 5.1: Cells $\tau_{I,J}$ for $\mathcal{X} = \mathbb{P}^2$

Theorems & Definitions (31)

• Remark
• Definition 2.2
• Remark
• Definition 3.1
• Example 3.2
• Example 3.3
• Proposition 4.1
• Lemma 4.2
• proof
• Proposition 4.3