The focus of my doctoral work has been answering the question: Can lower-dimensional effective gravitational theories be found in a higher-dimensional theory with a non-compact transverse space? To answer this question this thesis is divided into two parts. First, I explore supergravity solutions on warped product manifolds and argue that they correspond to solutions of a modified Laplacian. I pay special attention to Type III † solutions, or solutions characterized by the presence of non-constant transverse zero modes, and emphasize that these are the only higher dimensional solutions corresponding to localized sources that yield effectively lower-dimensional physics when the transverse space has infinite volume. Second, I derive the lower dimensional effective field theory about such backgrounds. I discover that
these effective field theories have covert symmetry breaking, spontaneous breaking of gauge symmetry which only appears at quartic order. I show this explicitly for D = d + 1 scalar electrodynamics with any boundary condition that corresponds to a non-constant transverse zero mode. The mathematical prerequisite for both of these conclusions is Sturm–Liouville theory with precise manipulations of Green’s
formula. To support this I derive the fundamental conclusions of Sturm–Liouville theory for a restricted class of operator, that is the Laplacian, which relaxes some requirements for permissible boundary conditions of Sturm–Liouville theory. The answer to my focal question is: Yes; however, the lower-dimensional theory has novel corrections which were previously unexplored, and further research is indicated.