The Swampland program aims to distinguish effective quantum field theories that can potentially arise within UV-complete quantum gravity theories (“the Landscape”) from those that cannot (“the Swampland”). The most well-established such criterion is that theories of quantum gravity never have exact global symmetries. I will discuss some generalizations and consequences of this idea. In particular, I will discuss work in progress (with Heidenreich, McNamara, Montero, Rudelius, and Valenzuela) on how axion fields often play a crucial role in banishing global symmetries from a theory, with potential phenomenological implications.
