On the topological character of metric-affine Lovelock Lagrangians in critical dimensions

In this paper we prove that the $k$-th order metric-affine Lovelock Lagrangian is not a total derivative in the critical dimension $n=2k$ in the presence of non-trivial non-metricity. We use a bottom-up approach, starting with the study of the simplest cases, Einstein-Palatini in two dimensions and Gauss-Bonnet-Palatini in four dimensions, and focus then on the critical Lovelock Lagrangian of arbitrary order. The two-dimensional Einstein-Palatini case is solved completely and the most general solution is provided. For the Gauss-Bonnet case, we first give a particular configuration that violates at least one of the equations of motion and then show explicitly that the theory is not a pure boundary term. Finally, we make a similar analysis for the $k$-th order critical Lovelock Lagrangian, proving that the equation of the coframe is identically satisfied, while the one of the connection only holds for some configurations. In addition to this, we provide some families of non-trivial solutions.