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 l ≤ 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 l = 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. (author)