A comment on Metric vs Metric-Affine Gravity

Abstract: We consider the sum of the Einstein-Hilbert action and a Pontryagin density (PD) in arbitrary even dimension $D$. All curvatures are functions of independent affine (torsionless) connections only. In arbitrary 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 Levi-Civita, 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.