Monopole operators and Hilbert series of Coulomb branches of 3 d $ \mathcal{N} $ = 4 gauge theories

This paper addresses a long standing problem - to identify the chiral ring and moduli space (i.e. as an algebraic variety) on the Coulomb branch of an $ \mathcal{N} $ = 4 superconformal field theory in 2+1 dimensions. Previous techniques involved a computation of the metric on the moduli space and/or mirror symmetry. These methods are limited to sufficiently small moduli spaces, with enough symmetry, or to Higgs branches of sufficiently small gauge theories. We introduce a simple formula for the Hilbert series of the Coulomb branch, which applies to any good or ugly three-dimensional $ \mathcal{N} $ = 4 gauge theory. The formula counts monopole operators which are dressed by classical operators, the Casimir invariants of the residual gauge group that is left unbroken by the magnetic flux. We apply our formula to several classes of gauge theories. Along the way we make various tests of mirror symmetry, successfully comparing the Hilbert series of the Coulomb branch with the Hilbert series of the Higgs branch of the mirror theory