A Critical Behaviour of Anomalous Currents, Electric-Magnetic   Universality and CFT_4
  We discuss several aspects of superconformal field theories in four dimensions (CFT_4), in the context of electric-magnetic duality. We analyse the behaviour of anomalous currents under RG flow to a conformal fixed point in N=1, D=4 supersymmetric gauge theories. We prove that the anomalous dimension of the Konishi current is related to the slope of the beta function at the critical point. We extend the duality map to the (nonchiral) Konishi current. As a byproduct we compute the slope of the beta function in the strong coupling regime. We note that the OPE of $T_{\mu\nu}$ with itself does not close, but mixes with a special additional operator $\Sigma$ which in general is the Konishi current. We discuss the implications of this fact in generic interacting conformal theories. In particular, a SCFT_4 seems to be naturally equipped with a privileged off-critical deformation $\Sigma$ and this allows us to argue that electric-magnetic duality can be extended to a neighborhood of the critical point. We also stress that in SCFT_4 there are two central charges, c and c', associated with the stress tensor and $\Sigma$, respectively; c and c' allow us to count both the vector multiplet and the matter multiplet effective degrees of freedom of the theory. 
D. Anselmi, M. Grisaru, A. Johansen