  These notes are an expanded version of lectures given at the 2022 TASI summer
school in Boulder, Colorado. One goal of these lecture notes is to (partially)
bridge the gap between what one learns in typical introductory quantum field
theory classes and what one needs to understand to follow modern developments
in particle theory beyond the Standard Model. Topics covered include global and
gauge symmetries, charge quantization, instantons, chiral anomalies, the Strong
CP problem, axion models from 4d and from higher dimensions, the expected
absence of global symmetries in quantum gravity, and some phenomenological
implications thereof. If these topics seem to be at best loosely related, I
hope that reading the notes will convince you otherwise. Recent developments in
quantum field theory have shown that ordinary gauge theories exhibit a much
wider range of (generalized) global symmetries than were previously understood,
while recent work in quantum gravity has provided sharper arguments that global
symmetries should not exist. This sets up an interesting tension, the
resolution of which can have implications for particle physics in the real
world. Axion physics is one setting in which these ideas can guide
phenomenology. The TASI audience comprised particle phenomenology students whom
I hoped to convince of the importance of learning more about quantum field
theory and quantum gravity. These notes may also be of interest to formal
theory readers seeking closer connections to real-world particle physics.
