Linearised E Theory in Teleparallel Gravity

• Linearised E theory is a teleparallel formulation of extended gravity that organizes its dynamics into a first-order differential chain complex with inherent gauge symmetry.
• The approach uses homotopy transfer to derive second-order equations by algebraically eliminating auxiliary fields, emphasizing its first-order foundation.
• By integrating torsion modules, dualisation maps, and local rotations, the framework contrasts conventional two-derivative Einstein gravity with manifest symmetry and algebraic clarity.

Searching arXiv for recent and relevant papers on linearised E-theory, especially teleparallel formulations and E11-related linearised gravity. Linearised E-theory is the linearised dynamics of the teleparallel version of extended geometry, including gravity, formulated by a chain complex whose differential provides the dynamics. In this formulation the basic differential contains only terms of order $0$ and $1$ in derivatives, while second derivatives arise from homotopy transfer, equivalently from eliminating fields with algebraic equations of motion. The construction is stated for a finite-dimensional structure group $G$, with exceptional groups $E_n$ as the usual examples, and it is designed so that symmetry under local rotations becomes manifest rather than emerging from a special combination of tensorial terms (Cederwall et al., 2023).

1. Formal setting and field content

The starting point is a complex $\mathcal C$ built from $G$-modules arranged by ghost number. The coordinate module is $V=R(-\lambda)$, higher reducibility modules are denoted $V'$, the linearised generalised vielbein sits in $\mathfrak g\oplus \mathbf 1$, the torsion sector is $\Theta$, and the local-rotation sector is an involutory subalgebra $1$0. Below ghost number $1$1 one recovers the dual modules of torsion, vielbein, and ghosts (Cederwall et al., 2023).

The central part of the complex is

$1$2

together with the dual chain

$1$3

The linearised generalised vielbein is written in components as

$1$4

The torsion sector consists of a covariantly projected “big” torsion $1$5 and a “small” torsion $1$6, while the map $1$7 lands in the antisymmetric projection $1$8 containing the local subalgebra $1$9 (Cederwall et al., 2023).

Ghost number	Module	Interpretation
$G$0	$G$1	ghosts
$G$2	$G$3	linearised generalised vielbein
$G$4	$G$5	torsion
$G$6	$G$7	local rotations
$G$8	dual modules	dual torsion, vielbein, ghosts

This organisation is significant because the kinematics, gauge structure, torsion field strengths, Bianchi identities, and dual modules are all encoded as arrows of a single complex rather than introduced as separate ingredients (Cederwall et al., 2023).

2. Differential structure and teleparallel consistency

The full differential is

$G$9

Here $E_n$0 is the derivative part, with one derivative, and $E_n$1 is algebraic. The central maps are given explicitly by (Cederwall et al., 2023)

$E_n$2

$E_n$3

and

$E_n$4

In these expressions, $E_n$5 are the representation matrices of $E_n$6 on $E_n$7, $E_n$8 is the projector selecting the covariant torsion submodule inside $E_n$9, and the braces $\mathcal C$0 denote projection onto the leading antisymmetric piece in $\mathcal C$1, which contains $\mathcal C$2 (Cederwall et al., 2023).

The algebraic part of the differential contains the dualisation map $\mathcal C$3, and its defining consistency condition is

$\mathcal C$4

This condition fixes $\mathcal C$5 uniquely, up to trivial redefinitions, once an invariant splitting $\mathcal C$6 is chosen. In the simplest gravity-like case one finds

$\mathcal C$7

where $\mathcal C$8 and $\mathcal C$9 are simple endomorphisms of the torsion module (Cederwall et al., 2023).

The conceptual consequence is that the linearised dynamics are governed by a first-order differential constrained by a strict algebraic compatibility condition. This is the sense in which the teleparallel complex replaces the usual two-derivative starting point.

3. Linearised dynamics and emergence of second-order equations

The linearised BV action is

$G$0

with $G$1 the sum over fields at different ghost numbers and $G$2 the natural pairing. The physical sector sits in ghost number $G$3 for $G$4 and in ghost number $G$5 for $G$6. Restricting to these terms gives (Cederwall et al., 2023)

$G$7

where $G$8 is the linearised torsion.

Varying with respect to $G$9 yields an algebraic equation of motion,

$V=R(-\lambda)$0

Substituting back gives the second-order action

$V=R(-\lambda)$1

The same result is expressed homologically by homotopy transfer. With strong homotopy retraction data

$V=R(-\lambda)$2

together with inclusion and projection maps $V=R(-\lambda)$3, the transferred differential is

$V=R(-\lambda)$4

Thus the second-order operator is not fundamental; it is the derived bracket produced by transferring the first-order complex to the cohomology of $V=R(-\lambda)$5 (Cederwall et al., 2023).

A common misunderstanding is to treat the second-order equations as the primary formulation. In the teleparallel complex this is not the case: the first-order differential $V=R(-\lambda)$6 is primary, and the two-derivative dynamics arise only after algebraic elimination or homotopy transfer (Cederwall et al., 2023).

4. Relation to conventional linearised formulations

The conventional non-teleparallel linearisation writes a two-derivative equation such as $V=R(-\lambda)$7, or its generalised counterpart. In that description gauge invariance is not manifest at the outset and instead appears through a cancellation among several terms in the two-derivative Lagrangian. By contrast, in the teleparallel complex the entire gauge symmetry and the Bianchi identities are encoded directly in the first-order differential $V=R(-\lambda)$8 (Cederwall et al., 2023).

The same source identifies four structural consequences. First, the dynamical operator $V=R(-\lambda)$9 is purely algebraic. Second, it remains undeformed in the interacting BV theory. Third, homotopy transfer packages the second-order structure into the single bracket $V'$0. Fourth, $V'$1-invariance becomes manifest because local rotations appear as exact differentials in the complex (Cederwall et al., 2023).

This stands in informative contrast with standard linearised gravity. On an Einstein background satisfying $V'$2, the linearised Einstein equations can be organised into the sequence

$V'$3

where $V'$4 and $V'$5 is the linearised Einstein operator constructed from the Calabi operator (Eastwood, 2022). This suggests a close formal affinity between linearised E-theory and gauge-complex formulations of linearised gravity, although the teleparallel complex is distinguished by its explicit use of torsion modules and algebraic dualisation.

5. Teleparallel Ehlers geometry as the basic example

The paper’s worked example is teleparallel Ehlers geometry, obtained when the extended coordinate module is the adjoint module of a finite-dimensional simple Lie group. In this case one takes $V'$6 to be the adjoint of $V'$7, so that (Cederwall et al., 2023)

$V'$8

The projector tensor becomes

$V'$9

and the linearised torsion is

$\mathfrak g\oplus \mathbf 1$0

with additional small torsion

$\mathfrak g\oplus \mathbf 1$1

The corresponding Bianchi identity in the leading antisymmetric piece is

$\mathfrak g\oplus \mathbf 1$2

and projection onto the adjoint of the involutory subalgebra $\mathfrak g\oplus \mathbf 1$3 produces the final arrow $\mathfrak g\oplus \mathbf 1$4 (Cederwall et al., 2023).

The dualisation operator $\mathfrak g\oplus \mathbf 1$5 is then determined by imposing

$\mathfrak g\oplus \mathbf 1$6

on the adjoint $\mathfrak g\oplus \mathbf 1$7. For $\mathfrak g\oplus \mathbf 1$8 and $\mathfrak g\oplus \mathbf 1$9, the torsion module decomposes into

$\Theta$0

of $\Theta$1, with $\Theta$2-eigenvalues

$\Theta$3

respectively. Inserting the resulting $\Theta$4 into

$\Theta$5

reproduces exactly the linearised teleparallel Ehlers action (Cederwall et al., 2023).

This example is important because it shows that the abstract complex is not merely formal. It can be diagonalised representation by representation, and the resulting algebraic dualisation reproduces a concrete linearised action.

6. Relation to dual gravity and low-level $\Theta$6

A broader algebraic context is provided by the $\Theta$7 approach to gravity. In four dimensions, after deleting node $\Theta$8 of the $\Theta$9 Dynkin diagram, the lowest levels under $1$00 contain the vielbein $1$01, scalars $1$02, vectors $1$03, and the dual graviton $1$04 satisfying $1$05. The corresponding generalised spacetime includes the coordinates $1$06 and $1$07 (West, 2014).

In that setting the gravity sector is governed by a first-order equation

$1$08

with

$1$09

After truncating to ordinary spacetime, setting all other fields to zero, and linearising

$1$10

one obtains

$1$11

and hence a linearised first-order relation involving both the usual graviton $1$12 and the dual graviton $1$13 (West, 2014).

The dual graviton is then eliminated by taking derivatives, tracing, symmetrising, and fixing the residual Lorentz gauge. The result is the familiar linearised Einstein equation

$1$14

This suggests a broader pattern: both teleparallel E-theory and low-level $1$15 gravity begin from first-order structures with auxiliary sectors and recover ordinary linearised Einstein dynamics only after eliminating non-propagating or dual variables (West, 2014).

The same paper emphasises that a full non-linear completion remains open: one must understand how to truncate the infinite generalised spacetime, how to treat the level-zero local subgroup systematically, and how to derive a manifestly $1$16-invariant action or set of field equations. Linearised E-theory should therefore be viewed as a well-defined linear regime inside a larger programme of extended geometric and algebraic formulations of gravity (West, 2014).