Supercurrent anomalies in 4d SCFTs

We use holographic renormalization of minimal $\mathcalN=2$ gauged supergravity in order to derive the general form of the quantum Ward identities for 3d $\mathcalN=2$ and 4d $\mathcalN=1$ superconformal theories on general curved backgrounds, including an arbitrary fermionic source for the supercurrent. The Ward identities for 4d $\mathcalN=1$ theories contain both bosonic and fermionic global anomalies, which we determine explicitly up to quadratic order in the supercurrent source. The Ward identities we derive apply to any superconformal theory, independently of whether it admits a holographic dual, except for the specific values of the $a$ and $c$ anomaly coefficients, which are equal due to our starting point of a two-derivative bulk supergravity theory. In the case of 4d $\mathcalN=1$ superconformal theories, we show that the fermionic anomalies lead to an anomalous transformation of the supercurrent under rigid supersymmetry on backgrounds admitting Killing spinors, even if all anomalies are numerically zero on such backgrounds. The anomalous transformation of the supercurrent under rigid supersymmetry leads to an obstruction to the $Q$-exactness of the stress tensor in supersymmetric vacua, and may have implications for the applicability of localization techniques. We use this obstruction to the $Q$-exactness of the stress tensor in order to resolve a number of apparent paradoxes relating to the supersymmetric Casimir energy, the BPS condition for supsersymmetric vacua, and the compatibility of holographic renormalization with supersymmetry, that were presented in the literature.