Consistency of eight-dimensional supergravities: Anomalies, Lattices and Counterterms

We reexamine the question of quantum consistency of supergravities in eight dimensions. Theories with both 32 and 16 supercharges suffer from the anomalies under the action of their respective discrete modular groups. In maximal supergravity the anomaly cancellation requires a surprising modification of the Chern-Simons couplings. In minimally supersymmetric theory coupled to Yang-Mills multiples of rank $l$ with the moduli space given by $\text{SO}(2,l)/ (\text{U}(1) \times \text{SO}(l))$, the existence of a counterterm together with the requirement that its poles and zeros correspond to the gauge symmetry enhancement imposes nontrivial constraints on the lattice. The counterterms needed for anomaly cancellation for all cases, that are believed to lead to consistent theories of quantum gravity ($l = 2,10,18$), are discussed