Generalized Symmetries, Noether currents, and the ABJ anomaly

We will study how the presence of generalized symmetries in a theory is related, from a local QFT point of view to the emergence of Haag duality violating (HDV) operator classes. In this context, we explain that if the non-local operators associated with the generalized symmetries transform under the action of a continuous global symmetry group then there cannot be a well-defined Noether current for such a group. We will show how this result can be used to re-derive Weinberg-Witten theorem within local QFT, where we will explicitly discuss the example of linearized gravitons. Furthermore, generalized symmetries that are charged under a continuous global symmetry, must be generated by a continuous of non-commuting dual HDV classes. If the HDV classes form a non-compact group, then we find that a theory with an unbroken non-compact higher-form symmetry is necessarily massless and free. However, if the HDV group is compact, the presence of charged classes seems compatible with an interacting QFT. In this case, the prototypical example is the chiral symmetry in QFT violated by the presence of the Adler-Bell-Jackiw anomaly. We will show how to understand the anomaly as a full abelian U(1) symmetry generated by a non-gauge-invariant Noether current. This entails the construction of the U(1) symmetry transformations for arbitrary, topologically non-trivial, spacetime regions, and showing they can transform the non-local charges.