On the number of terms in the Lovelock products

Xavier Lachaume

doi:10.1140/epjc/s10052-019-6776-6

Abstract

In this short note we wonder about the explicit expression of the expanding of the p-th Lovelock product. We use the 1990s’ works of S. A. Fulling et al. on the symmetries of the Riemann tensor, and we show that the number of independent scalars appearing in this expanding is equal to the number of Young diagrams with all row lengths even in the decomposition of the p-th plethysm of the Young diagram representing the symmetries of the Riemann tensor.

Highlights

Lovelock theories are a set of modified gravity theories that can be seen as generalisations of General Relativity (GR) in higher dimension
The non-vanishing term of highest-degree coincides to the Gauss–Bonnet–Chern scalar of the space-time manifold, exhibits a promising relation with geometry
The number of even row length diagrams will be the number of independent scalars which can be made by contractions of p copies of Rμρσν

Summary

Introduction

Lovelock theories are a set of modified gravity theories that can be seen as generalisations of General Relativity (GR) in higher dimension They could have interesting cosmological implications (see [1,2,3,4,5]) or connections with string/Mtheories in which higher-order curvature terms appear naturally (see [6]). They can be represented in the form of an action by a sum of scalar contractions of multiple copies of the Riemann curvature tensor. We are the first one to make an explicit connection with Lovelock theories

Notations

Young diagrams

Plethysms of the Riemann tensor

Conclusion