A comment on Metric vs Metric-Affine Gravity

TL;DR

This paper demonstrates that adding a Pontryagin density to the Einstein-Hilbert action in any even dimension results in a theory where the affine connection becomes Levi-Civita, making the metric and affine theories equivalent, contrary to previous theorems.

Contribution

It provides a counterexample showing that metric and metric-affine models can be equivalent beyond Lovelock theories by including Pontryagin densities in arbitrary even dimensions.

Findings

Pontryagin density is a covariant divergence of Chern-Simons current in any even dimension.

The connection field equations enforce Levi-Civita connection, linking metric and affine formulations.

The result challenges the belief that only Lovelock theories have metric-affine equivalence.

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.