Global anomalies in 8d supergravity

We study gauge and gravitational anomalies of fermions and 2-form fields on eight-dimensional spin manifolds. Possible global gauge anomalies are classified by spin bordism groups Ω9spinBG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\varOmega}_9^{\mathrm{spin}}(BG) $$\end{document} which we determine by spectral sequence techniques, and we also identify their explicit generator manifolds. It turns out that a fermion in the adjoint representation of any simple Lie group, and a gravitino in 8dN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 1 supergravity theory, have anomalies. We discuss how a 2-form field, which also appears in supergravity, produces anomalies which cancel against these fermion anomalies in a certain class of supergravity theories. In another class of theories, the anomaly of the gravitino is not cancelled by the 2-form field, but by topological degrees of freedom. It gives a restriction on the topology of spacetime manifolds which is not visible at the level of differential-form analysis.