Anomalies involving the space of couplings and the Zamolodchikov metric

The anomaly polynomial of a theory can involve not only curvature two-forms of the flavor symmetry background but also two-forms on the space of coupling constants. As an example, we point out that there is a mixed anomaly between the R-symmetry and the topology of the space of exactly marginal couplings of class S theories. Using supersymmetry, we translate this anomaly to the K\"ahler class of the Zamolodchikov metric. We compare the result against a holographic computation in the large N limit. In an appendix, we explain how to obtain the bosonic components in the supergravity completion of the curvature squared terms using compatibility with the topological twist.