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,e7ande8, but no construction for so(2 N+ 1) , f₄ and g₂ is known. In this paper, we show that the theories for f₄ and so(2 N+ 1) have a global gauge anomaly associated to π_d=8, while g₂ does not have it. We argue that the anomaly associated to π_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.