Aspects of effective potentials in gauge theories.

In this thesis we present several results concerning effective potentials in gauge theories. Firstly we calculate the effective potential up to one loop of weakly coupled large N N = 4 super Yang-Mills theory with gauge group SU(N) on S[3] x S[1] at low temperature and near-critical chemical potentials. We show that the theory contains a metastable phase for chemical potentials below a certain temperature- dependent value. We then give a review of Vilkovisky's formulation of the effective action, and use this method to calculate the effective potential of scalar electrodynamics with background VEVs for the scalar field and gauge field. Finally we describe the classical vacuum structure of a class of N = 1 gauge theories, which may be used to calculate the exact effective potential using the Dijkgraaf-Vafa conjecture