Inverse Hamiltonian reduction for affine W-algebras in type A

We give a geometric proof of inverse Hamiltonian reduction for all affine W-algebras in type A at generic level, a certain embedding of the affine W-algebra corresponding to an arbitrary nilpotent in $\mathfrak{gl}_N$ into that corresponding to a larger nilpotent with respect to the closure order on orbits, tensored with an auxiliary algebra of free fields. We proceed by constructing strict chiral quantizations of equivariant Slodowy slices, sheaves of $\hbar$-adic vertex algebras on the arc spaces of the slices, and then localizing them on quasi-Darboux open sets. We also provide a generalization for the Drinfeld-Sokolov reduction of arbitrary vertex algebra objects in the Kazhdan-Lusztig category.