Palatini Gauss-Bonnet theory

TL;DR

Palatini Gauss-Bonnet theories extend Chern-Simons-like structures to even dimensions by treating a GL$(2n)$ connection and an internal metric as independent variables, yielding a topological-like action in general. In 2D the theory reduces to a constrained BF system with only first-class constraints and no local degrees of freedom, modulo a novel hidden gauge symmetry; in 4D the dynamics become nontrivial, with a covariant formulation and a detailed Hamiltonian analysis revealing nine physical degrees of freedom and additional hidden gauge symmetries. A key methodological feature is a random-rank analysis of the constraint matrix to identify generic (maximum-rank) gauge structures and to count degrees of freedom, including subtle reducibilities. The results illuminate the gauge structure and constraint dynamics of higher-curvature Palatini theories and point to interesting boundary dynamics and further generalizations in related gauge groups.

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.}