Lovelock Terms and BRST Cohomology

TL;DR

This paper classifies Lovelock terms, polynomial scalar densities in the Riemann tensor, using BRST cohomology, and finds that no new generalized terms arise beyond the known ones.

Contribution

It reformulates the classification of Lovelock terms as a BRST cohomological problem and proves that no additional generalized Lovelock terms exist beyond the standard ones.

Findings

All generalized Lovelock terms are equivalent to the usual ones.

Allowing covariant derivatives does not produce new Lovelock structures.

The BRST cohomological approach provides a complete algebraic classification.

Abstract

Lovelock terms are polynomial scalar densities in the Riemann curvature tensor that have the remarkable property that their Euler-Lagrange derivatives contain derivatives of the metric of order not higher than two (while generic polynomial scalar densities lead to Euler-Lagrange derivatives with derivatives of the metric of order four). A characteristic feature of Lovelock terms is that their first nonvanishing term in the expansion of the metric around flat space is a total derivative. In this paper, we investigate generalized Lovelock terms defined as polynomial scalar densities in the Riemann curvature tensor and its covariant derivatives (of arbitrarily high but finite order) such that their first nonvanishing term in the expansion of the metric around flat space is a total derivative. This is done by reformulating the problem as a BRST cohomological one and by using cohomological tools. We determine all the generalized Lovelock terms. We find, in fact, that the class of nontrivial generalized Lovelock terms contains only the usual ones. Allowing covariant derivatives of the Riemann tensor does not lead to new structure. Our work provides a novel algebraic understanding of the Lovelock terms in the context of BRST cohomology.