Abstract
We consider dimensional reduction of the eleven-dimensional supergravity to less than four dimensions. The three-dimensional E 8(+8)/ SO(16) nonlinear sigma model is derived by direct dimensional reduction from eleven dimensions. In two dimensions we explicitly check that the Matzner-Misner-type SL(2, R) symmetry, together with the E 8, satisfies the generating relations of E 9 under the generalized Geroch compatibility (hypersurface-orthogonality) condition. We further show that an extra SL(2, R) symmetry, which is newly present upon reduction to one dimension, extends the symmetry algebra to a real form of E 10. The new SL(2, R acts on certain plane wave solutions propagating at the speed of light. To show that this SL(2, R) cannot be expressed in terms of the old E 9 but truly enlarges the symmetry, we compactify the final two dimensions on a two-torus and confirm that it changes the conformal structure of this two-torus.
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