Abstract
We construct the non-linear realisation of the semi-direct product of E11 and its vector representation in eleven dimensions and find the dynamical equations it predicts at low levels. These equations are completely determined by the non-linear realisation and when restricted to contain only the usual fields of supergravity and the usual space-time we find precisely the equations of motion of eleven dimensional supergravity. This paper extends the results announced in arXiv:1512.01644 and in particular it contains the contributions to the equations of motion that involve derivatives with respect to the level one generalised coordinates.
Highlights
Quite some time ago it was conjectured that the low energy effective action for strings and branes is the non-linear realisation of the semi-direct product of E11 and its vector (l1) representation, denoted E11 ⊗s l1 [1,2]
In this paper we have constructed the dynamics that follow from the non-linear realisation of E11 ⊗s l1 in eleven dimensions for the low level fields and generalised coordinates
The result is unique and when we truncate it to contain only the usual fields of supergravity, that is, the graviton and the three form, and take only the usual coordinates of spacetime we find the equations of motion of eleven dimensional supergravity
Summary
Quite some time ago it was conjectured that the low energy effective action for strings and branes is the non-linear realisation of the semi-direct product of E11 and its vector (l1) representation, denoted E11 ⊗s l1 [1,2]. A more systematic approach was used to constructing the equations of motion of the E11 ⊗s l1 non-linear realisation in eleven [3] and four [4] dimensions by including both the higher level generalised coordinates and local symmetries in Ic(E11) These papers did find the equations of motion of the form fields but found only partial results for the gravity equation. Section three derives the variations of the Cartan form under the symmetries of the non-linear realisation and in particular discusses an important subtilty associated with the fixing of the group element of the non-linear realisation using its local symmetry Using these results in section four we find the equations of motion for the three form and gravity and show that they vary into each other. Transformations of equation (1.3) they change as VE → h−1VE h + h−1dh and Vl → h−1Vlh (1.6)
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