Chern-Weil theorem, Lovelock Lagrangians in critical dimensions and boundary terms in gravity actions

TL;DR

This paper translates the Chern-Weil theorem into tensorial language for Lorentz symmetry, deriving vector densities whose divergences relate to Euler densities, with implications for Lovelock gravity and boundary terms in gravitational actions.

Contribution

It introduces a tensorial formulation of the Chern-Weil theorem, constructs vector densities for Euler and Lovelock densities, and clarifies boundary term structures in gravity actions.

Findings

Euler densities can be expressed as divergences of vector densities in critical dimensions

Constructed vector densities ensure well-posed variational principles in Lovelock gravity

Established tensorial translation of the Chern-Weil theorem for Lorentz symmetry

Abstract

In this paper we show how to translate into tensorial language the Chern-Weil theorem for the Lorentz symmetry, which equates the difference of the Euler densities of two manifolds to the exterior derivative of a transgression form. For doing so we need to introduce an auxiliary, hybrid, manifold whose geometry we construct explicitely. This allows us to find the vector density, constructed out of spacetime quantities only, whose divergence is the exterior derivative of the transgression form. As a consequence we can show how the Einstein-Hilbert, Gauss-Bonnet and, in general, the Euler scalar densities can be written as the divergences of genuine vector densities in the critical dimensions $D=2,4$, etc. As Lovelock gravity is a dimensional continuation of Euler densities, these results are of relevance for Gauss-Bonnet and, in general, Lovelock gravity. Indeed, these vectors which can be called generalized Katz vectors ensure, in particular, a well-posed Dirichlet variational principle.