We analyse the very-extended Kac-Moody algebras as representations in terms of certain A d subalgebras and find the generators at low levels. Our results for low levels agree precisely with the bosonic field content of the theories containing gravity, forms and scalars which upon reduction to three dimensions can be described by a non-linear realisation. We explain how the Dynkin diagrams of the very-extended algebras encode information about the field content and generalised T-duality transformations.

We investigate the generalised diffeomorphisms in M-theory, which are gauge transformations unifying diffeomorphisms and tensor gauge transformations. After giving an En(n)-covariant description of the gauge transformations and their commutators, we show that the gauge algebra is infinitely reducible, i.e., the tower of ghosts for ghosts is infinite. The Jacobiator of generalised diffeomorphisms gives such a reducibility transformation. We give a concrete description of the ghost structure, and demonstrate that the infinite sums give the correct (regularised) number of degrees of freedom. The ghost towers belong to the sequences of rep- resentations previously observed appearing in tensor hierarchies and Borcherds algebras. All calculations rely on the section condition, which we reformulate as a linear condition on the cotangent directions. The analysis holds for n < 8. At n = 8, where the dual gravity field becomes relevant, the natural guess for the gauge parameter and its reducibility still yields the correct counting of gauge parameters.Comment: 24 pp., plain tex, 1 figure. v2: minor changes, including a few added ref

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We study the hidden symmetries of the fermionic sector of D = 11 supergravity, and the rôle of K(E 10 ) as a generalised 'R-symmetry'. We find a consistent model of a massless spinning particle on an E 10 /K(E 10 ) coset manifold whose dynamics can be mapped onto the fermionic and bosonic dynamics of D = 11 supergravity in the near space-like singularity limit. This E 10 -invariant superparticle dynamics might provide the basis of a new definition of M-theory, and might describe the 'de-emergence' of spacetime near a cosmological singularity. Eleven-dimensional supergravity (SUGRA 11 ) [1] is believed to be the low-energy limit of the elusive 'M-theory', which is, hopefully, a unified framework encompassing the various known string theories. Understanding the symmetries of SUGRA 11 is therefore important for reaching a satisfactory formulation of M-theory. Many years ago it was found that the toroidal dimensional reduction of SUGRA 11 to lower dimensions leads to the emergence of unexpected ('hidden') symmetry groups, notably E 7 in the reduction to four noncompactified spacetime dimensions [2], E 8 in the reduction to D = 3 [2-5], and the affine Kac-Moody group E 9 in the reduction to D = 2 [6,7]. It was also conjectured [8] that the hyperbolic Kac-Moody group E 10 might appear when reducing SUGRA 11 to only one (time-like) dimension.Recently, the consideration, à la Belinskii, Khalatnikov and Lifshitz [9], of the near space-like singularity limit 1 of generic inhomogeneous bosonic eleven-dimensional supergravity solutions has uncovered some striking evidence for the hidden rôle of E 10 [10,11]. Ref.[11] related the gradient expansion * Corresponding author.E-mail address: axel.kleinschmidt@aei.mpg.de (A. Kleinschmidt). 1 This limit can also be viewed as a small tension limit, α → ∞.(∂ x ∂ t ), which organises the near space-like singularity limit [12], to an algebraic expansion in the height of positive roots of E 10 . A main conjecture of [11] was the existence of a correspondence between the time evolution, around any given spatial point x, of the supergravity bosonic fields g (11) MN (t, x), A (11) MNP (t, x), together with their infinite towers of spatial gradients, on the one hand, and the dynamics of a structureless massless particle on the infinite-dimensional coset space E 10 /K(E 10 ) on the other hand. Here, K(E 10 ) is the maximal compact subgroup of E 10 . Further evidence for the rôle of the one-dimensional non-linear sigma model E 10 /K(E 10 ) in M-theory was provided in [13][14][15][16].An earlier and conceptually different proposal aiming at capturing hidden symmetries of M-theory, and based on the very-extended Kac-Moody group E 11 , was made in [17,18] and further developed in [19][20][21]. A proposal combining the ideas of [18] and [11] was put forward in [22][23][24].In this Letter, we extend the bosonic coset construction of [11] to the full supergravity theory by including fermionic variables; more specifically, we provide evidence for the existence of a correspondence between th...