Moduli spaces of gauge theories in 3 dimensions.

The objective of this thesis is to study the moduli spaces of pairs of mirror theories in 3 dimensions with N = 4. The original conjecture of 3d mirror symmetry was motivated by the fact that in these pairs of theories the Higgs and Coulomb branches are swapped. After a brief introduction to supersymmetry we will first focus on the Higgs branch. This will be investigated through the Hilbert series and the plethystic program. The methods used for the Higgs branch are very well known in literature, more difficult is the case of the Coulomb branch since it receives quantum corrections. We will explain how it is parametrized in term of monopole operators and having both Higgs and Coulomb branches for theories with different gauge groups we will be able to show how mirror symmetry works in the case of ADE theories. We will show in which cases these Yang- Mills vacua are equivalent to one instanton moduli spaces.