Coulomb Branch Operators and Mirror Symmetry in Three Dimensions

We develop new techniques for computing exact correlation functions of a class of local operators, including certain monopole operators, in three-dimensional 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 ≤ 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 N=2 theories decorated by BPS 't Hooft-Wilson loops.