Anomaly-free supergravity theories remain anomaly-free after dimensional reduction

We show that the anomaly after compactification of a supergravity theory coupled to Yang-Mills matter is usually given by an integral of the original anomaly over the compact space, as long as there are no isometries for the compact space. This means that a supergravity theory, whose anomaly vanishes identically (i.e., without the addition of local counter terms to the action), will remain anomaly-free after compactification to any lower dimension, subject to some restrictions on self-dual antisymmetric tensors. We next consider the case where the original anomaly cancels by the Green-Schwarz mechanism. In this case, again subject to the restrictions on self-dual antisymmetric tensors, the anomaly will still cancel after compactification to any lower dimension D > 2, provided that: (1) There are no U(1) gauge groups after compactification. (2) There exists a three-form field strength H such that dH = (TrR₀² + kTrF₀²), or that the compact space is chosen such that (TrR₀² + kTrF₀²) = 0.