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), e(6), e(7) and e(8), but no construction for so(2N+1), f(4) and g(2) is known. In this paper, we show that the theories for f(4) and so(2N+1) have a global gauge anomaly associated to pi(d=8), while g(2) does not have it. We argue that the anomaly associated to pi(d) 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 Green Schwarz mechanism.