Broken current anomalous dimensions, conformal manifolds and RG flows

We consider deformations of a conformal field theory that explicitly break some global symmetries of the theory. If the deformed theory is still a conformal field theory, one can exploit the constraints put by conformal symmetry to compute broken currents anomalous dimensions. We consider several instances of this scenario, using field theory techniques and also holographic ones, where necessary. Field theoretical methods suffice to discuss examples of symmetry-breaking deformations of the $O(N )$ model in $d=4-\epsilon$ dimensions. Holography is instrumental, instead, for computing current anomalous dimensions in $\beta$-deformed superconformal field theories, and in a class of supersymmetric RG flows at large $N$.