Yang-Mills Matrix Theory

In this thesis, we discuss bosonic and supersymmetric Yang-Mills matrix models with compact semi-simple gauge group. We begin by finding convergence properties for the models. In the supersymmetric case, we show that the partition function converges when D = 4, 6 and 10, and that correlation functions of degree k < kc = 2(D− 3) are convergent independently of the group. In the bosonic case we show that the partition function is convergent when D ≥ Dc, and that correlation functions of degree k < kc are convergent, and calculate Dc and kc for each group. We move on to consider the supersymmetric theories in both their Yang-Mills and cohomological formulations. Specialising to the case of SU(N) with large N , we find all quantities which are invariant under the supercharges. Finally we apply the deformation method of Moore, Nekrasov and Shatashvili directly to the supersymmetric Yang-Mills model with D = 4. We find a deformation of the action which generates mass terms for all the matrix fields whilst preserving some supersymmetry. This allows us to rigorously integrate over a BRST quartet and arrive at the well known formula of MNS. Thesis submitted for the Degree of Doctor of Philosophy in the University of Oxford June 2001