E10 symmetry in one-dimensional supergravity Original Research Article

We consider dimensional reduction of the eleven-dimensional supergravity to less than four dimensions. The three-dimensional E8(+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 E8, satisfies the generating relations of E9 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 E10. 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 E9 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.