We discuss some general aspects of renormalization group flows in four dimensions. Every such flow can be reinterpreted in terms of a spontaneously broken conformal symmetry.We analyze in detail the consequences of trace anomalies for the effective action of the Nambu-Goldstone boson of broken conformal symmetry. While the c-anomaly is algebraically trivial, the a-anomaly is "non-Abelian," and leads to a positive-definite universal contribution to the S-matrix element of 2 → 2 dilaton scattering. Unitarity of the Smatrix results in a monotonically decreasing function that interpolates between the Euler anomalies in the ultraviolet and the infrared, thereby establishing the a-theorem.

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We give a complete geometric description of conformal anomalies in arbitrary, (necessarily even) dimension. They fall into two distinct classes: the first, based on Weyl invariants that vanish at integer dimensions, arises from finite -and hence scale-free -contributions to the effective gravitational action through a mechanism analogous to that of the (gauge field) chiral anomaly. Like the latter, it is unique and proportional to a topological term, the Euler density of the dimension, thereby preserving scale invariance. The contributions of the second class, requiring introduction of a scale through regularization, are correlated to all local conformal scalar polynomials involving powers of the Weyl tensor and its derivatives; their number increases rapidly with dimension.Explicit illustrations in dimensions 2, 4 and 6 are provided.Conformal (Weyl) anomalies have a long history (see for example [1][2][3]) and are still being studied (a partial list is in [4]), not least because of their many applications (e.g., [5][6][7][8][9]). However, their underlying geometric basis has not been systematized nor extended to general dimensions. It has also not been appreciated that there are in fact two distinct types of anomalies (one of which 1 On leave from the Weizmann Institute, Rehovot, Israel

Abstract. Using the relation between diffeomorphisms in the bulk and Weyl transformations on the boundary, we study the Weyl transformation properties of the bulk metric on-shell and of the boundary action. We obtain a universal formula for one of the classes of trace anomalies in any even dimension in terms of the parameters of the gravity action.