8d gauge anomalies and the topological Green-Schwarz mechanism

A bstract String theory provides us with 8d supersymmetric gauge theory with gauge algebras su(N),so(2N),sp(N),e₆,e₇0.5em and0.5em e₈ s u N , so 2 N , s p N , e 6 , e 7 and e 8 , but no construction for so(2N+1) so 2 N + 1 , f₄ f 4 and g₂ g 2 is known. In this paper, we show that the theories for f₄ f 4 and so(2N+1) so 2 N + 1 have a global gauge anomaly associated to π d =8 , while g₂ g 2 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) s p N have a subtler gauge anomaly, which we suggest should be canceled by a topological analogue of the Green-Schwarz mechanism.