Global anomalies in 8d supergravity

TL;DR

The authors classify global gauge and gravitational anomalies for eight-dimensional $\mathcal{N}=1$ supergravity by computing spin bordism groups $\Omega^{\text{spin}}_9(BG)$ via AHSS and Adams spectral sequences. They identify a universal adjoint-fermion anomaly and a gravitino anomaly, and demonstrate how a dynamical 2-form field can cancel these anomalies in many string-realizable theories, while certain models require topological (Wu-structure) degrees of freedom for cancellation. The work details explicit generator manifolds and shows how 2-form fields plus gauge/gravity CS terms can realize Green–Schwarz-type cancellations across various gauge groups, including $SU(n)$, $Sp(n)$, $G_2$, $F_4$, and $E_{6,7,8}$. In cases where 2-form cancellation is insufficient, the authors illuminate a topological mechanism involving a $\mathbb{Z}_2$ 3-form field that enforces constraints on spacetime topology, linking M-theory compactifications and Wu structures to anomaly cancellation and leading to a boundary TQFT description. Overall, the paper provides a comprehensive higher-dimensional anomaly taxonomy and clarifies the roles of form fields and topology in consistent 8d supergravity theories.

Abstract

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 $Ω^{\text{spin}}_9(BG)$ 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 $8d$ $\mathcal{N}=1$ 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.