For quantum field theories that flow between ultraviolet and infrared fixed points, central functions, defined from two-point correlators of the stress tensor and conserved currents, interpolate between central charges of the UV and IR critical theories. We develop techniques that allow one to calculate the flows of the central charges and that of the Euler trace anomaly coefficient in a general N=1 supersymmetric gauge theory. Exact, explicit formulas for SU (N c ) gauge theories in the conformal window are given and analysed. The Euler anomaly coefficient always satisfies the inequality a U V − a IR > 0. This is new evidence in strongly coupled theories that this quantity satisfies a four-dimensional analogue of the c-theorem, supporting the idea of irreversibility of the RG flow. Various other implications are discussed.

The relation between the trace and R-current anomalies in supersymmetric theories implies that the U(1) R F 2 , U(1) R and U(1) 3 R anomalies which are matched in studies of N = 1 Seiberg duality satisfy positivity constraints. Some constraints are rigorous and others conjectured as four-dimensional generalizations of the Zamolodchikov c-theorem. These constraints are tested in a large number of N = 1 supersymmetric gauge theories in the non-Abelian Coulomb phase, and they are satisfied in all renormalizable models with unique anomaly-free R-current, including those with accidental symmetry. Most striking is the fact that the flow of the Euler anomaly coefficient, a U V − a IR , is always positive, as conjectured by Cardy.