  In arXiv:0905.3629 we described a new class of N=2 topological amplitudes
that depends both on vector and hypermultiplet moduli. Here we find that this
class is actually a particular case of much more general topological amplitudes
which appear at higher loops in heterotic string theory compactified on K3 x
T^2. We analyze their effective field theory interpretation and derive
particular (first order) differential equations as a consequence of
supersymmetry Ward identities and the 1/2-BPS nature of the corresponding
effective action terms. In string theory the latter get modified due to
anomalous world-sheet boundary contributions, generalizing in a non-trivial way
the familiar holomorphic and harmonicity anomalies studied in the past. We
prove by direct computation that the subclass of topological amplitudes studied
in arXiv:0905.3629 forms a closed set under these anomaly equations and that
these equations are integrable.
