8d gauge anomalies and the topological Green-Schwarz mechanism

String theory provides us with 8d supersymmetric gauge theory with gauge algebras su(N), so(2N), sp(N), e6, e7 and e8, but no construction for so(2N+1), f4 and g2 is known. In this paper, we show that the theories for f4 and so(2N+1) have a global gauge anomaly associated to πd=8, while g2 does not have it. We argue that the anomaly associated to πd in d-dimensional gauge theories cannot be canceled by topological degrees of freedom in general. We also show that the theories for sp(N) have a subtler gauge anomaly, which we suggest should be canceled by a topological analogue of the GreenSchwarz mechanism.