Here we list a number of general identities for the completely antisymmetric volume tensor in
dimensions. The most useful are those involving the tensor product (including, as needed, contractions over
indices), of the volume tensor with itself [114]:
Start by writing Equation (360) as
The last thing we need is the variation of the determinant of the metric, since it enters directly in the
integrals of the actions. Treating the metric as a matrix, and “normalizing” the
by dividing by
its one independent component, the determinant is given by
http://www.livingreviews.org/lrr-2007-1 | ![]() This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License. Problems/comments to |