Moduli Spaces of N = 4, d = 3. Quiver Gauge Theories and Mirror Symmetry

In this thesis we study the structure of moduli spaces for N = 4 supersymmetric quiver gauge theories in d = 2 + 1 spacetime dimensions, which consist of Hyperk ahler cones. Such moduli spaces have two dierent branches, named Higgs Branch and Coulomb Branch, joined at the origin, which in turn corresponds to a superconformal xed point of the renormalization group ow. In this thesis, the standard procedure for computing the Higgs Branch via the Hyperk ahler quotient is reviewed. Furthermore, a novel approach (introduced for the rst time in [17]) to compute the Coulomb Branch is explained carefully. For this class of gauge theories there exist a conjectured Mirror Symmetry: a duality which swaps the moduli spaces' branches of dual theories. Along the way we provide some tests of such a symmetry. Applying this procedure, some new computation of such spaces are made, and some new mirror couples are conjectured.