What bordism-theoretic anomaly cancellation can do for U

We perform a bordism computation to show that the $E_{7(7)}(\mathbb{R})$ U-duality symmetry of 4d $\mathcal{N}=8$ supergravity could have an anomaly invisible to perturbative methods; then we show that this anomaly is trivial. We compute the relevant bordism group using the Adams and Atiyah-Hirzebruch spectral sequences, and we show the anomaly vanishes by computing $\eta$-invariants on the Wu manifold, which generates the bordism group.