8d gauge anomalies and the topological Green-Schwarz mechanism

String theory provides us with 8d supersymmetric gauge theory with gauge algebras $\mathfrak{s}\mathfrak{u}(N)$, $\mathfrak{s}\mathfrak{o}(2N)$, $\mathfrak{s}\mathfrak{p}(N)$, ${\mathfrak{e}}_6$, ${\mathfrak{e}}_7$ and ${\mathfrak{e}}_8$, but no construction for $\mathfrak{s}\mathfrak{o}(2N+1)$, ${\mathfrak{f}}_4$ and ${\mathfrak{g}}_2$ is known. In this paper, we show that the theories for ${\mathfrak{f}}_4$ and $\mathfrak{s}\mathfrak{o}(2N+1)$ have a global gauge anomaly associated to ${\pi }_{d=8}$, while ${\mathfrak{g}}_2$ 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 $\mathfrak{s}\mathfrak{p}(N)$ have a subtler gauge anomaly, which we suggest should be canceled by a topological analogue of the Green-Schwarz mechanism.