Twisting and Turning in Six Dimensions

Abstract: This thesis investigates certain aspects of a six-dimensional quantum theory known as (2,0) theory. This theory is maximally supersymmetric and conformal, making it the most symmetric higher dimensional quantum theory known. It has resisted an explicit construction as a quantum field theory yet its existence can be inferred from string theory. These properties suggests that an understanding of the theory will create a deeper understanding of the foundations of both. In the first part of the thesis an explicit formulation of the non-interacting ver- sion of the theory is investigated on space-time manifolds that are circle fibrations. The circle fibration geometry enables a compactification to a five dimensional su- persymmetric Yang-Mills theory. A unique extension to an interacting theory is found and conjectured to be the compactification of the interacting theory in six dimensions. The second part of the thesis concerns the topological twisting of the free theory in six dimensions. A space-time manifold which is a product of a four-dimensional and a two-dimensional part is considered. This setup has recently been proposed as an explaination for the conjectured correspondence between four dimensional gauge theory and two-dimensional conformal field theory known as the AGT corre- spondence. We perform the twisting and subsequent compactification on the two- dimensional manifold of the free tensor multiplet in Minkowski signature to avoid the problems associated with the definition of (2,0) theory on Euclidean manifolds. With the same choice of supercharge as in the usually preferred Euclidean scenario we conclude that there is no stress tensor which exhibits the topological properties previously found in similar theories.