Exploring Holographic Chaos and Complexity

In principle, holography provides a well-defined non-perturbative formulation of quantum gravity, but to really use it to address questions about the nature of spacetime, we need to understand the emergence of the bulk spacetime from the dual field theory description. A number of field theory properties are believed to be dual to aspects of the geometry, ranging from simple two-point correlation functions, known to be approximately dual to the length of geodesics, to the quantum complexity, which is conjectured to be dual to either the size of a maximal volume slice of the bulk, or to the action of a patch of the bulk spacetime. In this thesis, we use the duality between correlation functions and geodesics (and to a lesser extent, that between entanglement entropy and bulk surfaces proposed by Ryu and Takayanagi) to extend a holographic analysis of quantum chaos initiated by Shenker and Stanford. We then explore the more speculative holographic complexity conjectures, analysing the divergence of complexity in general and testing the conjectures in a number of specific spacetimes. For certain cases, we obtain qualitatively different results for the two conjectures, raising questions for future research