This extensive, very comprehensive and well-written article discusses properties of the Coulomb branch of three dimensional gauge theories with eight supercharges, \(\mathcal{N}=4\) supersymmetry. The Coulomb branch is the moduli space of supersymmetric vacua parametrized by vacuum expectation values (vevs) of a triplet of scalar fields. The key strategy is an “abelianization map” that embeds the Poisson algebra of holomorphic functions on the Coulomb branch into a larger algebra. It maps the vev of a monopole operator of the full theory to a linear combination of abelian monopole operators in the low-energy abelian gauge theory.
The article first gives a good introduction to the generalities of three dimensional \(\mathcal{N}=4\) theories and an subsequently the abelian Coulomb branches in such theories are discussed where the setup is generalized from SQED to a more general theory in Section 3. The key part are the non-abelian gauge theories discussed in Section 4 with their metric on the Coulomb branch and non-abelian monopole operators, where also higher-dimensional generalizations are touched. Finally, the last two sections deal with specific applications like SQCD in Section 5 and Linear Quivers of Unitary Groups in Section 6, where also Flavour symmetries and abelian coordinates are included. Interesting views are given from string theory: monopole scattering in type IIB string theories. The appendices give further aspects of singular monopoles as complex symplectic manifolds where the Poisson bracket is defined in terms of the scattering matrix and equivariant integrals and monopoles operators of higher charges.