On the Localization Equations of Topologically Twisted N=4 Super Yang-Mills Theory in Five Dimensions

In recent articles, topologically twisted N=4 supersymmetric Yang-Mills theory on a four-manifold of the form V=W x R^+ or V=W x I were considered. W here is a Riemannian three-manifold, and a suitable set of boundary conditions apply to the endpoints of I (or R^+). In the special case where W=S^3, spherically symmetric solutions where obtained to the localization equations. For large interval lengths, these consist of pairwise occurring (non gauge-equivalent) solutions, which then coincide for a certain critical interval length, only to disappear if it decreases below this critical value. Only for the instance were the interval length is of critical value was an exact analytical solution obtained. The only feasible explanation for this is that there exist a tunneling between the solutions in one solution-pair as one goes to five dimensions. This will be shown to be the case in this thesis. A five-dimensional version of the previously mentioned theory on R x S^3 x I is considered, and the localization equations of this theory obtained. An analytical expression of this five-dimensional supersymmetric field configuration has not been possible to obtain, similarly to the case in four dimensions, but the solution is instead obtained as a series expansion in terms of an infinitesimal parameter $\varepsilon$ stating how much the solutions differ from the exactly solvable static case for critical interval length in four dimensions, where we have stationary solutions in five dimensions as well