I will present a new approach to study the RG flow in 3d N=4 gauge theories, based on an analysis of the Coulomb branch of vacua. The Coulomb branch is described as a complex algebraic variety and important information about the strongly coupled fixed points of the theory can be extracted from the study of its singularities. I will use this framework to study the fixed points of USp(2N) gauge theories with fundamental matter, revealing some surprising features at low amount of matter.

I will discuss the partition function of 4d N=1 theories on Hopf surfaces. These are diffeomorphic to S^1 x S^3 and the partition function provides a path integral realization of the supersymmetric index. Its large S^1 limit exhibits a universal behaviour associated to the existence of a Casimir energy. I will argue that, contrarily to the non-supersymmetric case, this Casimir energy is a physical (non-ambiguous) quantity in supersymmetric theories.

We study half-BPS Wilson loops in D = 3 N =4 gauge theories using matrix models obtained from localization techniques. The infrared CFTs of the N=4 theories are subject to 3-dimensional mirror symmetry, which exchanges the Higgs and Coulomb branches of vacua of dual theories. Recently progress have been made in understanding the mapping of BPS Wilson loops under mirror symmetry in abelian theories. Our aim is to understand the operators dual to half-BPS Wilson loops in non-abelian theories. We propose a matrix model for the mirror loops by implementing mirror symmetry directly in the matrix model and we verify the mapping of loop operators by computing explicitly the Wilson loops and mirror loops in non-abelian linear quiver theories. We discuss the possible gauge theory operators that would lead to the matrix model we found. Our results are nicely related to the brane realization of linear quivers in IIB string theory.