Dimensional-reduction anomaly

In a wide class of D-dimensional spacetimes which are direct or semi-direct sums of a $\left(D-n\right)$-dimensional space and an n-dimensional homogeneous “internal” space, a field can be decomposed into modes. As a result of this mode decomposition, the main objects which characterize the free quantum field, such as Green functions and heat kernels, can effectively be reduced to objects in a $\left(D-n\right)$-dimensional spacetime with an external dilaton field. We study the problem of the dimensional reduction of the effective action for such spacetimes. While before renormalization the original D-dimensional effective action can be presented as a “sum over modes” of $\left(D-n\right)$-dimensional effective actions, this property is violated after renormalization. We calculate the corresponding anomalous terms explicitly, illustrating the effect with some simple examples.