Quantum D-module reduction to GIT quotients (informal conjecture)

Determine and prove that the T-equivariant quantum D-module QDM_T(X), when pulled back to the chart Spec(C[z][[C_{i,N}^\vee]]) corresponding to a chamber C_i in the T-ample cone, becomes (after specified modifications and graded completions) the quantum D-module of the associated smooth GIT quotient Y_i = X//_i T.

Background

The authors construct a global Kähler moduli space M from the chamber decomposition of the T-ample cone C_T(X) and argue that QDM_T(X) forms a sheaf over a formal subscheme of M×A^1_z. The conjecture posits that near each cusp corresponding to a chamber C_i (and GIT quotient Y_i), the sheaf restricts—after suitable modification and completion—to the quantum D-module QDM(Y_i).

This is an abstract D-module counterpart of the Fourier-duality picture, integrating shift (Seidel) operators with Novikov variables and aiming to globalize quantum cohomology across variation of GIT. The subsequent section formulates a more concrete reduction conjecture at the level of solutions (Givental cones).