| This thesis has two primary aims. One is to explore the limits on popular models of axion inflation due to generic constraints. The constraints that we study give some indication of what types of phenomenology are possible and expected from the perspective of a complete theory of quantum gravity. Specifically, we will use the weak gravity conjecture to constrain models of natural inflation, and we will use tunneling phenomena to constrain models of axion monodromy. These phenomenological goals are paramount in the pursuit of an unbroken connection between top-down quantum gravitational analysis and bottom-up experimental data. Although we will not fully bridge such a gap here, we do produce some additional planks to be used in such a construction.;The second aim is to explore the nature of the constraining principles themselves. Ultimately we would like to understand better the nature of quantum gravity by thorough investigation of the weak gravity conjecture itself, and we work towards such a goal in this work. Additionally, we would like to better understand the nature and practice of semiclassical physics. Semiclassical methods in field theory, especially in the context of tunneling, depend on Euclideanization in order for the path integral to be easily interpreted. However, only in simple cases, such as vacuum decay, are Euclidean methods well founded in terms of more basic semiclassical methods such as WKB. The success of Euclidean methods in these cases as well as its failures in more complicated settings are studied in detail. Ultimately, Euclidean techniques are deemed insufficient to handle the semiclassical analysis of general systems and an alternative, Picard-Lefschetz theory, is proposed and explored. |
