Fourier duality for quantum volumes under symplectic reduction

Establish that the T-equivariant quantum volume Π_X^{eq}(−[ω]−λ⋅μ) of a Hamiltonian T-space X and the quantum volume Π_{X//_t T}(−[ω_red]) of the symplectic (or GIT) reduction X//_t T are related by Fourier transformation with respect to λ ∈ Lie(T) and t ∈ Lie(T)^*, where [ω_red] denotes the class of the reduced symplectic form and ω−λ⋅μ is the Duistermaat–Heckman form. Concretely, demonstrate a Fourier duality Π_X^{eq}(−[ω]−λ⋅μ) ↔ Π_{X//_t T}(−[ω_red]) consistent with the identifications of parameters and the reduction data.

Background

The paper introduces the notion of quantum volume Π_X(τ) as a cohomological integral built from the small J-function and the Γ-class, motivated by Givental’s path-integral heuristics for Floer fundamental cycles. In classical symplectic geometry, equivariant volume and reduced volume are connected via Fourier transformation (Duistermaat–Heckman). The conjecture proposes a quantum analogue, asserting that the T-equivariant quantum volume of X and the (non-equivariant) quantum volume of the reduction X//_t T are Fourier dual, with λ (equivariant parameter) dual to t (stability parameter).

This conjecture underlies many examples in the paper (e.g., toric mirror symmetry, Mellin–Barnes representations for projective spaces) and connects difference-module structures on QH_T(X) to differential-module structures on QH(X//T).