Analytic validity of toric equivariant Fourier integrals beyond the weak-Fano case

Ascertain whether, in the toric mirror setting with X not Fano, the Fourier integral representation for the equivariant quantum volume (or its generalizations with z-dependent mirror maps) defines an actual analytic function beyond the weak-Fano case; if not, characterize its status as an asymptotic series in z and determine precise analytical conditions or domains of validity.

Background

In the toric Fano case, ΠX^{eq}(τ) admits an oscillatory integral representation over (R{>0})^n involving the toric mirror W_τ(x) and equivariant parameters. The authors note that for non-Fano toric varieties a z-dependent mirror map may be required and raise doubts about the analytic meaning of the corresponding Fourier integral.

Clarifying the analytic status of these Fourier integrals—convergence, summability, or interpretation as formal asymptotic series—is necessary to extend the proposed Fourier duality framework beyond weak-Fano settings.