Abstract
The non-linear dual graviton equation of motion as well as the duality relation between the gravity and dual gravity fields are found in E theory by carrying out E11 variations of previously found equations of motion. As a result the equations of motion in E theory have now been found at the full non-linear level up to, and including, level three, which contains the dual graviton field. When truncated to contain fields at levels three and less, and the spacetime is restricted to be the familiar eleven dimensional space time, the equations are equivalent to those of eleven dimensional supergravity.
Highlights
It was conjectured that the non-linear realisation of the semi-direct product of E11 with its vector representation (l1), denoted E11 ⊗s l1, leads to a theory, called E theory, that contains the eleven dimensional supergravity theory [1,2]
For example reference [3], it has been shown that E theory leads to essentially unique equations of motion which are second order in derivatives for the graviton and form fields [4,5]
By varying the previously deduced six form equation of motion we have found dual gravity equation of motion that is second order in derivatives up to the presence of some very specific terms whose form is discussed in the paper
Summary
It was conjectured that the non-linear realisation of the semi-direct product of E11 with its vector representation (l1), denoted E11 ⊗s l1, leads to a theory, called E theory, that contains the eleven dimensional supergravity theory [1,2]. For example reference [3], it has been shown that E theory leads to essentially unique equations of motion which are second order in derivatives for the graviton and form fields [4,5] When these equations of motion are restricted to contain only the supergravity fields which live on the usual spacetime they are just those of eleven dimensional supergravity [3,4]. The gravity-dual gravity relations given in reference [1] did correctly propagate the gravity degrees of freedom this was at the linearised level and this relation was postulated rather than derived from the E11 ⊗s l1 non-linear realisation in that paper. At level zero this subalgebra is the Lorentz algebra and the variations at the level have the parameter Λa1a2a3 under which the Cartan forms transform as [4,5]
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