The manifolds Hp,q [Symbol appears here. To view, please open pdf attachment] are a family of non-compact hyperboloids carrying inhomogeneous Euclidean metrics. In supergravity they appear as an interesting class of dimensional reductions, related to the well known sphere reductions by a simple analytic continuation. The spectrum of lower dimensional modes in these backgrounds is still poorly understood. In this thesis, we construct the complete Pauli reduction of type IIA supergravity on H2,2 [Symbol appears here. To view, please open pdf attachment] . We carefully analyse the spectrum of gravitational waves in the resulting Salam-Sezgin background, and identify the boundary conditions needed to render these modes normalisable. We give these boundary conditions a codimension-2 braneworld interpretation. We then exhibit a supersymmetric braneworld geometry based on the NS5-brane. In the remainder of this thesis we apply holographic methods to the problem of the fractionalisation transition in condensed matter theory. We exhibit a phase transition between a superconducting and a fractionalised phase in a bottom-up Einstein-Maxwell-Dilaton theory, and discuss the importance of entropy scaling in achieving this.