Cancelling mod-2 anomalies by Green-Schwarz mechanism with $B_{\mu \nu }$

We study if and when mod-2 anomalies can be canceled by the Green-Schwarz mechanism with the introduction of an antisymmetric tensor field $B_{\mu \nu }$. As explicit examples, we examine $SU(2)$ and more general $Sp(n)$ gauge theories in four and eight dimensions. We find that the mod-2 anomalies of 8d $\mathcal{N}=1$ $Sp(n)$ gauge theory can be canceled, as expected from it having a string theory realization, while the mod-2 Witten anomaly of 4d $SU(2)$ and $Sp(n)$ gauge theory cannot be canceled in this manner.