Teleparallel Conformal Invariant Models induced by Kaluza-Klein Reduction

Abstract

We study the extensions of teleparallism in the Kaluza-Klein (KK) scenario by writing the analogous form to the torsion scalar $T_{\text{NGR}}$ in terms of the corresponding antisymmetric tensors, given by $T_{\text{NGR}} = a\,T_{ijk} \, T^{ijk} + b\,T_{ijk} \,T^{kji} + c\,T^{j}{}_{ji} \, T^{k}{}_{k}{}^{i}$, in the four-dimensional New General Relativity (NGR) with arbitrary coefficients $a$, $b$ and $c$. After the KK dimensional reduction, the Lagrangian in the Einstein-frame can be realized by taking $2a+b+c=0$ with the ghost-free condition $c\leq0$ for the one-parameter family of teleparallelism. We demonstrate that the pure conformal invariant gravity models can be constructed by the requirements of $2a+b=0$ and $c=0$. In particular, the torsion vector can be identified as the conformal gauge field, while the conformal gauge theory can be obtained by $2a+b+4c=0$ or $2a+b=0$, which is described on the Weyl-Cartan geometry $Y_4$ with the ghost-free conditions $2a+b+c>0$ and $c\neq0$. We also consider the weak field approximation and discuss the non-minimal coupled term of the scalar current and torsion vector. For the conformal invariant models with $2a+b=0$, we find that only the anti-symmetric tensor field is allowed rather than the symmetric one.