Coulomb Branch Operators and Mirror Symmetry in Three Dimensions

TL;DR

This paper introduces new localization techniques to compute exact correlation functions of Coulomb branch operators in 3D $ =4$ abelian gauge theories, providing insights into their structure and mirror symmetry.

Contribution

It develops a novel approach using hemisphere localization and shift operators to compute Coulomb branch correlators and tests mirror symmetry in three-dimensional theories.

Findings

Exact correlation functions of Coulomb branch operators are computed.

New derivations of monopole operator dimensions are provided.

Results connect 3D theories with 4D Schur indices.

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 ($n\leq 3$)-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.