A comment on metric vs metric-affine gravity

We consider the sum of the Einstein-Hilbert action and a Pontryagin density (PD) in arbitrary even dimension D >= 4. All curvatures are functions of independent affine (torsionless) connections only. In arbitrary even dimension, not only in D = 4n, these first order PD terms are shown to be covariant divergences of "Chern-Simons" currents. The field equation for the connection leads to it being LeviCivita, and to the metric and affine field equations being equivalent to the second order metric theory. This result is a counterexample to the theorem stating that purely metric and metric-affine models can only be equivalent for Lovelock theories. (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP(3).