The Gauss-Bonnet topological scalar in the Geometric Trinity of Gravity

TL;DR

This paper explores the Gauss-Bonnet topological scalar across various teleparallel gravity frameworks, explicitly expressing it in terms of torsion and non-metricity, and analyzes its implications for symmetry preservation in higher-order theories.

Contribution

It provides explicit expressions for the Gauss-Bonnet scalar in different teleparallel formalisms and examines its role in maintaining symmetries in higher-order gravity theories.

Findings

The number of invariant terms is counted and compared with effective field theory.

The Gauss-Bonnet invariant excludes some effective field theory terms.

Symmetry preservation in teleparallel theories is highly nontrivial.

Abstract

The Gauss-Bonnet topological scalar is presented in metric-teleparallel formalism as well as in the symmetric and general teleparallel formulations. In all of the aforementioned frameworks, the full expressions are provided explicitly in terms of torsion, non-metricity and Levi-Civita covariant derivative. The number of invariant terms of this form is counted and compared with the number which can appear in the corresponding effective field theory. Although the difference in this number is not very large, it is found that the Gauss-Bonnet invariant excludes some of the effective field theory terms. This result sheds new light on how General Relativity symmetries can be maintained at higher order in teleparallel theories: this fact appears to be highly nontrivial in the teleparallel formulation. The importance of the so-called ``pseudo-invariant'' theories like $f(T)$- and $f(T,T_\mathcal{G})$-gravity is further discussed in the context of teleparallel Gauss-Bonnet gravity.