  String theory provides us with 8d supersymmetric gauge theory with gauge
algebras $\mathfrak{su}(N)$, $\mathfrak{so}(2N)$, $\mathfrak{sp}(N)$,
$\mathfrak{e}_{6}$, $\mathfrak{e}_{7}$ and $\mathfrak{e}_{8}$, but no
construction for $\mathfrak{so}(2N{+}1)$, $\mathfrak{f}_4$ and $\mathfrak{g}_2$
is known. In this paper, we show that the theories for $\mathfrak{f}_4$ and
$\mathfrak{so}(2N{+}1)$ 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
$\mathfrak{sp}(N)$ have a subtler gauge anomaly, which we suggest should be
canceled by a topological analogue of the Green-Schwarz mechanism.
