Quantization commutes with reduction again: the quantum GIT conjecture I

For a compact monotone symplectic manifold $X$ with Hamiltonian action of a compact Lie group $G$ and smooth symplectic reduction, we relate its gauged $2$-dimensional $A$-model to the $A$-model of $X/\!/G$. This (long conjectured) result is parallel to the ($B$-model!) \emph{quantization commutes with reduction} theorem of Guillemin and Sternberg in quantum mechanics. Here, we spell out some of the precise statements, and outline the proof of equality for the spaces of states (quantum cohomology). We also indicate the way to some related results in the non-monotone case. Additional Floer theory details will be included in a follow-up paper.