Gravity, dual gravity and A1+++

Abstract

We construct the nonlinear realisation of the semidirect product of the very extended algebra [Formula: see text] and its vector representation. This theory has an infinite number of fields that depend on a space–time with an infinite number of coordinates. Discarding all except the lowest level field and coordinates the dynamics is just Einstein’s equation for the graviton field. We show that the gravity field is related to the dual graviton field by a duality relation and we also derive the equation of motion for the dual gravity field.

A master exceptional field theory

“…3 In [3] a set of first-order 'modulo equations' were proposed for the coset field V. These were constructed out of the Maurer-Cartan derivatives that transform into each other under K(E 11 ) and are generated from the matter duality equation F 4 = F 7 . The terminology of modulo equation means a (first-order) equation that is not gauge-invariant (for gauge parameters depending on eleven coordinates) but only holds up to certain gauge transformations that can be eliminated by passing to a higher-order equation [3,14,15]. For the graviton the gauge-invariant equations are second order, and for more complicated fields gauge-invariance requires differential equations of arbitrarily high order [3,14,15].…”

“…The terminology of modulo equation means a (first-order) equation that is not gauge-invariant (for gauge parameters depending on eleven coordinates) but only holds up to certain gauge transformations that can be eliminated by passing to a higher-order equation [3,14,15]. For the graviton the gauge-invariant equations are second order, and for more complicated fields gauge-invariance requires differential equations of arbitrarily high order [3,14,15]. The gauge transformations and the intrinsic multiplet structure of the whole K(E 11 )-multiplet of first-order modulo equations are not known to the best of our knowledge.…”

“…However, there are also examples where such a form exists without the requirement of complete reducibility. 15 In the following we shall assume that η IJ exists. Our investigations to date have not unveiled any submodule R(λ) so that T −1 might be irreducible, but at present (2.16) is an assumption and we do not have proof of the existence of η IJ whose existence is crucial for our model.…”

“…Note that since the Bianchi identity (8.63) is not the lowest weight component of a lowest weight E 11 representation, these higher-order integrability conditions cannot be organised in lowest weight representations of E 11 . As we discussed in section 8.1, the need for these higher order equations appears to be unrelated to the approach in [3,14,15].…”

“…One can also envisage performing a construction similar to the one of the present paper for theories without maximal supersymmetry. This would involve DFT itself, but also cases such as pure general relativity in D = 4 spacetime dimensions by replacing E 11 by other very-extended Kac-Moody algebras such as A +++ 1 [15,101]. For previous work on non-pure supergravity theories see for example [124,125].…”