The geometry and topology of quantum entanglement in holography

In this thesis I explore the connection between geometry and quantum entanglement, in the context of holographic duality, where entanglement entropies in a quantum field theory are associated with the areas of surfaces in a dual gravitational theory. The first chapter looks at a phase transition in such systems in finite size and at finite temperature, associated with the properties of minimal surfaces in a static black hole background. This is followed by the related problem of extremal surfaces in a spacetime describing the dynamical process of black hole formation, with a view towards understanding the connections between bulk locality and various field theory observables including entanglement entropy. The third chapter looks at the simple case of pure gravity in three spacetime dimensions, where I show how evaluating the entanglement entropy can be reduced to a simple algebraic calculation, and apply it to some interesting examples. Finally, the role played by topology of surfaces in a proposed derivation of a holographic entanglement entropy formula is investigated. This makes it clear what assumptions are required in order to reproduce the ‘homology constraint’, a topological condition necessary for consistency with field theory