Reduction conjecture for equivariant Gromov–Witten theory of GIT quotients

Let $X$ be a smooth projective variety with an algebraic action of a complex torus $T_{\mathbb C}$, and let $Y=X/\!/T_{\mathbb C}$ be a smooth GIT quotient with no orbifold singularities. Write $J_X(\tau,z)$ for the big $T$-equivariant $J$-function and $\kappa\colon H_T^*(X)\to H^*(Y)$ for the Kirwan map.

Using the equivariant shift operators $\widehat{\mathcal S}^{\beta}$, form the discrete Fourier transform

$$I_Y=\sum_{[\beta]}\kappa\!\left(\widehat{\mathcal S}^{-\beta}J_X\right)\widehat S^\beta,$$

where $[\beta]$ runs over equivariant curve classes modulo ordinary curve classes.

The Iritani–Sanda reduction conjecture predicts that this series is supported on the semigroup dual to the GIT chamber of $Y$ and that

$$zI_Y\in\mathcal L_Y,$$

where $\mathcal L_Y$ is the non-equivariant Givental cone of $Y$ (over the corresponding Novikov completion).

Thus a discrete Fourier transform of the equivariant genus-zero Gromov–Witten theory of $X$ should recover the genus-zero Gromov–Witten theory—and quantum $D$-module—of the quotient $Y$. The conjecture is verified in basic toric examples, but remains open in general.