We show that generalized symmetries cannot be charged under a continuous global symmetry having a Noether current. Further, only non-compact generalized symmetries can be charged under a continuous global symmetry. These results unravel topological obstructions to the existence of a Noether current in QFT. We discuss some examples, including the free massless graviton. We use these results to rederive Weinberg-Witten’s theorem within local QFT, generalizing it to massless particles in arbitrary dimensions and representations of the Lorentz group. We end up discussing the conditions for the existence of a Noether current, dynamical aspects of QFT’s with non-compact generalized symmetries, scale vs conformal invariance and the Coleman-Mandula theorem.
Based on [arXiv:2205.03412 [hep-th]]