E 10 and SO ( 9 , 9 ) invariant supergravity

We show that (massive) D = 10 type IIA supergravity possesses a hidden rigid D 9 ≡ SO(9, 9) symmetry and a hidden local SO(9) × 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 E 10 /K(E 10) non-linear σ-model to the level ℓ ≤ 2 sector in a decomposition of E 10 under its D 9 subalgebra. This decomposition is presented up to level ℓ = 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 DE 10 , associated to the latter, is shown to be a proper subalgebra of E 10 , in accord with the embedding of type I into type IIA supergravity. The corresponding decomposition of DE 10 under D 9 is presented up to level ℓ = 5.