Constraints on compressible metals through emergent symmetries and anomalies

I will summarize recent work showing that the language of emergent symmetries and anomalies provides a powerful framework for thinking about properties of non-Fermi liquids at low temperatures without reference to any particular model. I will argue that clean compressible metals necesssarily have a very non-trivial emergent symmetry and anomaly structure, likely associated with the existence of a Fermi surface. I will show how to understand Luttinger's theorem in a very general way using these ideas.