Teleparallelism in the algebraic approach to extended geometry

TL;DR

This paper develops a teleparallel framework within extended geometry using tensor hierarchy algebras, linking local symmetries, field strengths, and Bianchi identities to generalize gravity's teleparallel formulation.

Contribution

It extends the $L_infty$ structure to include physical fields and derives a dual gauge symmetry, generalizing Lorentz invariance in teleparallel gravity.

Findings

Field strengths as generalized torsion and Bianchi identities.

Identification of dual gauge symmetry as local invariance under a compact subgroup.

Application to models with finite-dimensional structure groups, including $E_5$.

Abstract

Extended geometry is based on an underlying tensor hierarchy algebra. We extend the previously considered $L_\infty$ structure of the local symmetries (the diffeomorphisms and their reducibility) to incorporate physical fields, field strengths and Bianchi identities, and identify these as elements of the tensor hierarchy algebra. The field strengths arise as generalised torsion, so the naturally occurring complex in the $L_\infty$ algebra is $\ldots\leftarrow$ torsion BI's $\leftarrow$ torsion $\leftarrow$ vielbein $\leftarrow$ diffeomorphism parameters $\leftarrow\ldots$ In order to obtain equations of motion, which are not in this complex, (pseudo-)actions, quadratic in torsion, are given for a large class of models. This requires considering the dual complex. We show how local invariance under the compact subgroup locally defined by a generalised metric arises as a "dual gauge symmetry" associated with a certain torsion Bianchi identity, generalising Lorentz invariance in the teleparallel formulation of gravity. The analysis is performed for a large class of finite-dimensional structure groups, with $E_5$ as a detailed example. The continuation to infinite-dimensional cases is discussed.