Structure of Lanczos-Lovelock Lagrangians in Critical Dimensions

The Lanczos-Lovelock models of gravity constitute the most general theories of gravity in D dimensions which satisfy (a) the principle of of equivalence, (b) the principle of general co-variance, and (c) have field equations involving derivatives of the metric tensor only up to second order. The mth order Lanczos-Lovelock Lagrangian is a polynomial of degree m in the curvature tensor. The field equations resulting from it become trivial in the critical dimension $D = 2m$ and the action itself can be written as the integral of an exterior derivative of an expression involving the vierbeins, in the differential form language. While these results are well known, there is some controversy in the literature as to whether the Lanczos-Lovelock Lagrangian itself can be expressed as a total divergence of quantities built only from the metric and its derivatives (without using the vierbeins) in $D = 2m$. We settle this issue by showing that this is indeed possible and provide an algorithm for its construction. In particular, we demonstrate that, in two dimensions, $R \sqrt{-g} = \partial_j R^j$ for a doublet of functions $R^j = (R^0,R^1)$ which depends only on the metric and its first derivatives. We explicitly construct families of such R^j -s in two dimensions. We also address related questions regarding the Gauss-Bonnet Lagrangian in $D = 4$. Finally, we demonstrate the relation between the Chern-Simons form and the mth order Lanczos-Lovelock Lagrangian.