Chaos, wormholes, and holography A tale of topology in quantum gravity

Combining quantum mechanics and general relativity is one of the most important open problems in theoretical physics. A formal approach towards quantizing gravity is given by the gravitational path integral. However, dealing with quantum fluctuations of spacetime itself is in general a very complicated task. In this thesis, I therefore study a simple model of two-dimensional gravity in Anti-de Sitter (AdS) space where it is possible to calculate exactly a class of such quantum fluctuations, so-called Euclidean wormholes, which correspond to geometries with a non-trivial topology. This computation relies on the holographic principle: the idea that AdS gravity is dual to some microscopic quantum theory without gravity, a conformal field theory (CFT), that lives in one dimension less. Surprisingly, the non-trivial surfaces appear to contain important information about this underlying quantum theory: They point to a deep connection between quantum gravity and quantum chaos. To be precise, the results of this thesis show that black holes in two-dimensional AdS space obey the rules of quantum chaos, and that the relevant quantum effects can be realized geometrically in the path integral. This is an important step towards understanding the quantum nature of black holes. Finally, I study the role of geometric phases in the AdS/CFT correspondence, and show that the modular Berry phase in the CFT is holographically dual to a particular geometric quantity in the AdS space, the so-called bulk symplectic form.