Giving a $KO$ to 8D Gauge Anomalies

Abstract: In \cite{Garcia-Etxebarria:2017crf}, it was found that the system of $k$ D7-branes probing an $O7^+$-plane suffers from an $\mathfrak{sp}(k)$ gauge anomaly when $k>1$. These authors then conjectured that this 8D $\mathcal{N}=1$ gauge theory couples to an 8D topological field theory (TFT) such that the total system is anomaly-free, thus acting as a "topological" Green-Schwarz mechanism. In this note, we construct such an 8D TFT and show that it indeed cancels the gauge anomaly. The key step is to engineer the relevant topological operators from D3-branes and fluxbranes placed infinitely far away from the stack of 7-branes. Such symmetry operators have topological vector bundles defined on them whose $KO/KSp$-homology classes play a role in the anomaly cancellation.