Abstract
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 ≤ 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 HS3 with half-BPS boundary conditions, where operator insertions within the hemisphere are represented by certain shift operators acting on the HS3 wavefunction. By gluing a pair of such wavefunctions, we obtain correlators on S3 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.
Highlights
N = 4 supersymmetry in three dimensions provides a rich middle ground between the availability of calculable supersymmetry-protected observables and nontrivial dynamics
N = 4 supersymmetry allows for various calculations of protected observables that led to the discovery of the duality and to various tests thereof, such as the match between the infrared metrics of the Coulomb and Higgs branches [3], scaling dimensions of monopole operators [4], various curved-space partition functions [5,6,7], expectation values of loop operators [8, 9], and the Hilbert series [10]
The theory has flavor symmetry GH × GC where GH acts on the hypermultiplet, while GC generally emerges in the IR and acts on the Coulomb branch
Summary
Our goal in the present paper is to provide new insights into the mirror symmetry duality and, more generally, into 3D N = 4 QFTs, by developing new techniques for calculating correlation functions of certain CBOs that include monopole operators. These techniques are related to the observation of [11, 12] that all N = 4 superconformal field theories (SCFTs) contain two one-dimensional topological sectors, one associated with the Higgs branch and one associated with the Coulomb branch. Having explicit descriptions of both the Higgs and Coulomb branch 1D sectors allows for more explicit tests of mirror symmetry, including a precise mapping between all half-BPS operators of the two theories
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