Marginal deformations and defect anomalies

We deform a defect conformal field theory by an exactly marginal bulk operator and we consider the dependence on the marginal coupling of flat and spherical defect expectation values. For odd-dimensional defects, we find a different qualitative behavior for the flat and spherical case, generalizing to arbitrary dimensions the line-circle anomaly of superconformal Wilson loops. In the even-dimensional case, on the other hand, we find a logarithmic divergence which can be related to a a-type anomaly coefficient. This coefficient, for defect theories, is not invariant on the conformal manifold and its dependence on the bulk coupling is controlled to all orders by the one-point function of the associated exactly marginal operator. In particular, our results imply a nontrivial dependence on the bulk coupling for the recently proposed defect C-function. We finally apply our general result to a few specific examples, including superconformal Wilson loops and Rényi entropy.