Coulomb branch operators and mirror symmetry in three dimensions

Abstract

Abstract We develop new techniques for computing exact correlation functions of a class of local operators, including certain monopole operators, in three-dimensional N=4 $$ \mathcal{N}=4 $$ 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 ≤ 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 $$ \mathcal{N}=2 $$ theories decorated by BPS ’t Hooft-Wilson loops.

Keywords

• Extended Supersymmetry
• Supersymmetric Gauge Theory
• Supersymmetry and Duality
• Conformal Field Theory