E_{10} Symmetry in One-dimensional Supergravity

TL;DR

This paper explores the emergence of an $E_{10}$ symmetry in one-dimensional supergravity obtained from eleven-dimensional supergravity, revealing new symmetries and their algebraic structures through dimensional reduction and compactification.

Contribution

It demonstrates how an extra $SL(2,R)$ symmetry extends the known $E_9$ symmetry to $E_{10}$ in one-dimensional supergravity, providing explicit checks and algebraic relations.

Findings

Identification of $E_{10}$ symmetry in 1D supergravity

Extension of $E_9$ to $E_{10}$ via new $SL(2,R)$ symmetry

Modification of the two-torus conformal structure upon reduction

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.