Spontaneously Broken Boosts in CFTs

Conformal Field Theories (CFTs) have rich dynamics in heavy states. We describe the constraints due to spontaneously broken boost and dilatation symmetries in such states. The spontaneously broken boost symmetries require the existence of new low-lying primaries whose scaling dimension gap, we argue, scales as O(1). We demonstrate these ideas in various states, including fluid, superfluid, mean field theory, and Fermi surface states. I will go through some of these examples, present the general argument and end with some remarks about the large charge limit in 2d and discuss a theory of a single compact boson with an arbitrary conformal anomaly (based on arXiv:2102.12583 [hep-th] with Zohar Komargodski, Mark Mezei and Avia Raviv-Moshe.)