Field Theories with a Twist

Abstract: This thesis deals with topological field theories derived from maximally supersymmetric gauge theories. The main part of this work is the three appended research papers, each of which investigates some different aspect of these theories. Leading up to these is the first part of the thesis which aims at introducing the above-mentioned concepts in such a fashion that the papers become legible to even those readers who are less familiar with notions such as topological field theory and N=4 Yang-Mills theory from the start. This introduction begins with the concept of symmetry, and both gauge- and supersymmetry are explained. This leads on to the introduction of maximally supersymmetric gauge theories, and then finally to outlining the possible ways to create topological field theories from these. The two last chapters of this thesis are dedicated to the theories featured in the appended papers. Papers I and II both deals with the geometric Langlands-twist of N=4 Yang-Mills theory and the five-dimensional analog of this. In paper I, spherically symmetric solutions to the localisation equations of both the four- and five-dimensional topological field theories are investigated, whereas paper II presents an explicit expression for the action of the five-dimensional theory. The final paper deals with a topological twisting of (2,0) theory, which was conjectured to be relevant in the understanding of the AGT-correspondence. However, this suggested twist is shown to not result in a topological field theory on a general background by showing that the stress-tensor of the theory cannot in general be both $Q$-exact and conserved.