  We develop new techniques for computing exact correlation functions of a
class of local operators, including certain monopole operators, in
three-dimensional $\mathcal{N} = 4$ abelian gauge theories that have
superconformal infrared limits. These operators are position-dependent linear
combinations of Coulomb branch operators. They form a one-dimensional
topological sector that encodes a deformation quantization of the Coulomb
branch chiral ring, and their correlation functions completely fix the ($n\leq
3$)-point functions of all half-BPS Coulomb branch operators. Using these
results, we provide new derivations of the conformal dimension of half-BPS
monopole operators as well as new and detailed tests of mirror symmetry. Our
main approach involves supersymmetric localization on a hemisphere $HS^3$ with
half-BPS boundary conditions, where operator insertions within the hemisphere
are represented by certain shift operators acting on the $HS^3$ wavefunction.
By gluing a pair of such wavefunctions, we obtain correlators on $S^3$ with an
arbitrary number of operator insertions. Finally, we show that our results can
be recovered by dimensionally reducing the Schur index of 4D $\mathcal{N} = 2$
theories decorated by BPS 't Hooft-Wilson loops.
