E₁₀ Symmetry in One-dimensional Supergravity

الأصل · EN

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.