Anomalies and Conformal Manifolds

We study conformal field theories with exactly marginal deformations. In even dimensions they have trace anomalies. In addition to the familiar ones I will discuss anomalies which appear in the presence of these deformations.
When the underlying conformal field theory is supersymmetric, they provide information about the conformal manifolds M of these theories. For N = (2, 2) theories in d = 2 and N = 2 theories in d = 4 we establish the relation between the sphere partition function and the Kahler potential of M.