Stability of D-brane geometries and a quantum check of AdS/CFT

The first part of this thesis explores the stability of non-supersymmetric constructions using D-branes and M-branes. Guided mainly by intuition developed using the correspondence between gauge theory and gravity, known as AdS/CFT, we propose a precise relationship between thermodynamic and dynamical stability of non-extremal branes. We verify the conjecture explicitly for non-extremal M2-branes with angular momentum in planes perpendicular to the world-volume, in the limit of many M2-branes where the supergravity approximation is reliable. Next, we explore the stability of near-horizon geometries of extremal branes which are product geometries of anti-de Sitter space and positively curved Einstein spaces. Our main motivation is to answer the question: Do non-supersymmetric stable vacua exist? We find that the answer is yes. But for Type IIA strings in the presence of D8-branes and for a non-supersymmetric open string theory with gauge group USp(32) we find that spherical compactifications are unstable.; The second part of this thesis explores AdS/CFT predictions beyond the classical level. Such checks are usually hard to carry out, at least in the absence of supersymmetry. We find an interesting test which yields a manifestly finite answer without using supersymmetry. It involves calculating the one-loop vacuum energy of a tachyon field in anti-de Sitter space with boundary conditions corresponding to the presence of a double-trace operator in the dual field theory. Such an operator can lead to a renormalization group flow between two different conformal field theories related to each other by a Legendre transformation in the large N limit. The calculation of the one-loop vacuum energy enables us to verify the holographic c-theorem one step beyond the classical supergravity approximation.