Palatini Gauss-Bonnet theory

TL;DR

This paper explores a class of polynomial gravity models in even dimensions, similar to Chern-Simons theories, analyzing their degrees of freedom and reformulations, especially in 2 and 4 dimensions.

Contribution

It introduces a polynomial class of models with a $GL(2n)$ connection and metric, analyzing their degrees of freedom and reformulations in various dimensions.

Findings

No local degrees of freedom in 2D, reformulated as a constrained BF model.

Presence of local degrees of freedom in higher dimensions, such as 4D.

Detailed analysis of 2D and 4D cases.

Abstract

We consider a class of models in even spacetime dimensions $2n$ which share many similarities with Chern-Simons theories in odd spacetime dimensions $2n+1$. The independent dynamical variables of these models are a $GL(2n)$-connection and a metric in internal space. The action is a polynomial of degree $n$ in the curvature of the connection, with indices saturated by means of the metric and the Levi-Civita tensor. We show that the theory has no local degree of freedom in $2$ spacetime dimensions ($n=1$), where it can be reformulated as a constrained $BF$ model, but that its dynamics is more intrincate in higher dimensions ($n>1$), where local degrees of freedom are present. We treat in detail the cases of $2$ and $4$ spacetime dimensions.}