Abstract
We construct a pseudo-Lagrangian that is invariant under rigid E11 and transforms as a density under E11 generalised diffeomorphisms. The gauge-invariance requires the use of a section condition studied in previous work on E11 exceptional field theory and the inclusion of constrained fields that transform in an indecomposable E11-representation together with the E11 coset fields. We show that, in combination with gauge-invariant and E11-invariant duality equations, this pseudo-Lagrangian reduces to the bosonic sector of non-linear eleven-dimensional supergravity for one choice of solution to the section condi- tion. For another choice, we reobtain the E8 exceptional field theory and conjecture that our pseudo-Lagrangian and duality equations produce all exceptional field theories with maximal supersymmetry in any dimension. We also describe how the theory entails non-linear equations for higher dual fields, including the dual graviton in eleven dimensions. Furthermore, we speculate on the relation to the E10 sigma model.
Highlights
It has been proposed by West [1,2,3] that it should be possible to write D = 11 supergravity in a way that utilises the infinite-dimensional Kac-Moody symmetry E11.1 The proposed construction involves a non-linear realisation of E11/K(E11) where K(E11) ⊂ E11 denotes a subgroup that generalises the eleven-dimensional Lorentz groupSO(1, 10) and will be defined in more detail below.2 The fields of the non-linear realisation depend on space-time coordinates zM that transform under E11 in an infinitedimensional highest weight representation [2] that we shall call R(Λ1) in this paper
The gauge-invariance requires the use of a section condition studied in previous work on E11 exceptional field theory and the inclusion of constrained fields that transform in an indecomposable E11-representation together with the E11 coset fields
In an effort to define an exceptional field theory for E11 we have recently proposed a non-linear set of first-order duality equations [13] that can be written as MIJ F I = ΩIJ F J, (1.1)
Summary
It depends on the E11 coset fields only through the e11 current, without the explicit appearance of the generalised metric M It is defined as a rigid E11-invariant completion of the total derivative of a constrained field transforming in an indecomposable representation together with the e11 current, as does the topological term in E9 ExFT [30]. Appendix E contains details of the GL(1) × E10 decomposition and remarks on the relation to the E10 sigma model Since this is a rather long paper, readers primarily interested in seeing how elevendimensional supergravity emerges from the proposed master exceptional field theory may focus on sections 2, 3, 6 and 7
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