Perturbative Aspects of Supersymmetric Yang-Mills Theory on T3

Licentiatavhandling, 2010

This thesis is concerned with perturbative aspects of supersymmetric Yang-Mills theory in a space-time of the form T3xR, in the regime where the gauge coupling is weak. Wave functions of vacuum states of super Yang-Mills theories are supported on the moduli space of flat connections, and we consider a semi-classical treatment of such theories, obtained by expanding the fields around some normalizable zero energy background field configuration. In particular, we consider the theories localized at a certain class of vacua corresponding to isolated points in the moduli space. In the limit where the gauge coupling vanishes, and the degrees of freedom of the theory decouple, the theory describes equal numbers of non-interacting massless bosonic and fermionic fields related by supersymmetry. The energy spectrum of the free theory is determined by the eigenvalues of the covariant derivative with respect to the isolated flat background connection. We discuss the construction of the free Hilbert space and compute the energy spectrum for all isolated vacua in simple gauge groups using Lie algebra theory. Subsequently, we consider the interacting theory (still in the weak coupling regime) for the case when the gauge group is SU(n)/Zn. We discuss the construction of the Hilbert space of this theory using perturbation theory in the coupling constant. We also consider the perturbative corrections to the energy spectrum to quadratic order and show that supersymmetry implies a non-trivial cancellation of UV divergences rendering these corrections finite.