E10 and SO(9,9) invariant supergravity

We show that (massive) D=10 type IIA supergravity possesses a hidden rigid SO(9,9) symmetry and a hidden local SO(9) x SO(9) symmetry upon dimensional reduction to one (time-like) dimension. We explicitly construct the associated locally supersymmetric Lagrangian in one dimension, and show that its bosonic sector, including the mass term, can be equivalently described by a truncation of an E10/K(E10) non-linear sigma-model to the level \ell<=2 sector in a decomposition of E10 under its so(9,9) subalgebra. This decomposition is presented up to level 10, and the even and odd level sectors are identified tentatively with the Neveu--Schwarz and Ramond sectors, respectively. Further truncation to the level \ell=0 sector yields a model related to the reduction of D=10 type I supergravity. The hyperbolic Kac--Moody algebra DE10, associated to the latter, is shown to be a proper subalgebra of E10, in accord with the embedding of type I into type IIA supergravity. The corresponding decomposition of DE10 under so(9,9) is presented up to level 5.Comment: 1+39 pages LaTeX2e, 2 figures, 2 tables, extended tables obtainable by downloading sourc

Einstein billiards and spatially homogeneous cosmological models

In this paper, we analyse the Einstein and Einstein-Maxwell billiards for all spatially homogeneous cosmological models corresponding to 3 and 4 dimensional real unimodular Lie algebras and provide the list of those models which are chaotic in the Belinskii, Khalatnikov and Lifschitz (BKL) limit. Through the billiard picture, we confirm that, in D=5 spacetime dimensions, chaos is present if off-diagonal metric elements are kept: the finite volume billiards can be identified with the fundamental Weyl chambers of hyperbolic Kac-Moody algebras. The most generic cases bring in the same algebras as in the inhomogeneous case, but other algebras appear through special initial conditions.Comment: 27 pages, 10 figures, additional possibility analysed in section 4.3, references added, typos correcte

E10 and Gauged Maximal Supergravity

We compare the dynamics of maximal three-dimensional gauged supergravity in appropriate truncations with the equations of motion that follow from a one-dimensional E10/K(E10) coset model at the first few levels. The constant embedding tensor, which describes gauge deformations and also constitutes an M-theoretic degree of freedom beyond eleven-dimensional supergravity, arises naturally as an integration constant of the geodesic model. In a detailed analysis, we find complete agreement at the lowest levels. At higher levels there appear mismatches, as in previous studies. We discuss the origin of these mismatches.Comment: 34 pages. v2: added references and typos corrected. Published versio

E10 and a "small tension expansion" of M Theory

A formal ``small tension'' expansion of D=11 supergravity near a spacelike singularity is shown to be equivalent, at least up to 30th order in height, to a null geodesic motion in the infinite dimensional coset space E10/K(E10) where K(E10) is the maximal compact subgroup of the hyperbolic Kac-Moody group E10(R). For the proof we make use of a novel decomposition of E10 into irreducible representations of its SL(10,R) subgroup. We explicitly show how to identify the first four rungs of the E10 coset fields with the values of geometric quantities constructed from D=11 supergravity fields and their spatial gradients taken at some comoving spatial point.Comment: 4 page

Oscillatory regime in the Multidimensional Homogeneous Cosmological Models Induced by a Vector Field

We show that in multidimensional gravity vector fields completely determine the structure and properties of singularity. It turns out that in the presence of a vector field the oscillatory regime exists in all spatial dimensions and for all homogeneous models. By analyzing the Hamiltonian equations we derive the Poincar\'e return map associated to the Kasner indexes and fix the rules according to which the Kasner vectors rotate. In correspondence to a 4-dimensional space time, the oscillatory regime here constructed overlap the usual Belinski-Khalatnikov-Liftshitz one.Comment: 9 pages, published on Classical and Quantum Gravit

Black brane solutions related to non-singular Kac-Moody algebras

A multidimensional gravitational model containing scalar fields and antisymmetric forms is considered. The manifold is chosen in the form M = M_0 x M_1 x ... x M_n, where M_i are Einstein spaces (i > 0). The sigma-model approach and exact solutions with intersecting composite branes (e.g., solutions with harmonic functions and black brane ones) with intersection rules related to non-singular Kac-Moody (KM) algebras (e.g. hyperbolic ones) are considered. Some examples of black brane solutions are presented, e.g., those corresponding to hyperbolic KM algebras: H_2(q,q) (q > 2), HA_2^(1) = A_2^{++} and to the Lorentzian KM algebra P_{10}.Comment: 16 pages, Late

E11, Borcherds algebras and maximal supergravity

The dynamical p-forms of torus reductions of maximal supergravity theory have been shown some time ago to possess remarkable algebraic structures. The set ("dynamical spectrum") of propagating p-forms has been described as a (truncation of a) real Borcherds superalgebra V Dthat is characterized concisely by a Cartan matrix which has been constructed explicitly for each spacetime dimension 11 ≥ D ≥ 3. In the equations of motion, each differential form of degree p is the coefficient of a (super-) group generator, which is itself of degree p for a specific gradation (the V-gradation). A slightly milder truncation of the Borcherds superalgebra enables one to predict also the "spectrum" of the non-dynamical (D-1) and D-forms. The maximal supergravity p-form spectra were reanalyzed more recently by truncation of the field spectrum of E 11to the p-forms that are relevant after reduction from 11 to D dimensions. We show in this paper how the Borcherds description can be systematically derived from the split ("maximally non compact") real form of E 11for D ≥ 1. This explains not only why both structures lead to the same propagating p-forms and their duals for p ≤ (D-2), but also why one obtains the same (D-1)-forms and "top" D-forms. The Borcherds symmetries V 2and V 1are new too. We also introduce and use the concept of a presentation of a Lie algebra that is covariant under a given subalgebra.SCOPUS: ar.jinfo:eu-repo/semantics/publishe