Higher order M theory corrections and the Kac-Moody algebra E10

TL;DR

Damour and Nicolai advance the program that M-theory dynamics is governed by a one-dimensional sigma-model on the infinite-dimensional coset $E_{10}/K(E_{10})$ by connecting leading higher-order corrections to negative imaginary roots of $E_{10}$. They derive a precise selection rule: curvature corrections of the form $R^M (DF)^N$ are compatible with $E_{10}$ only if $M+N=3k+1$, and show that the leading quartic corrections $R^4$ (and $R^7$, etc.) correspond to ${\mathfrak{sl}}_{10}$ singlets at levels $\ell=10$ and $\ell=20$, respectively. The analysis uses the BKL-like asymptotics and a cosmological billiard in the $E_{10}$ root space to predict the behavior near cosmological singularities and suggests a richer, more intricate dynamical structure than homogeneous ansätze. The work provides concrete algebraic tests and predictions for higher-order M-theory corrections, linking their structure to the deep representation theory of $E_{10}$ and offering a framework for constraining and understanding corrections beyond the two-derivative supergravity action.

Abstract

It has been conjectured that the classical dynamics of M theory is equivalent to a null geodesic motion in the infinite-dimensional coset space E10/K(E10) where K(E10) is the maximal compact subgroup of the hyperbolic Kac-Moody group E10. We here provide further evidence for this conjecture by showing that the leading higher order corrections, quartic in the curvature and related three-form dependent terms, correspond to negative imaginary roots of E10. The conjecture entails certain predictions for which higher order corrections are allowed: in particular corrections of type R^M (DF)^N are compatible with E10 only for M+N=3k+1. Furthermore, the leading parts of the R^4, R^7,... terms are predicted to be associated with singlets under the SL(10) decomposition of E10. Although singlets are extremely rare among the altogether 4,400,752,653 representations of SL(10) appearing in E10 up to level l \leq 28, there are indeed singlets at levels l=10 and l =20 which do match with the R^4 and the expected R^7 corrections. Our analysis indicates a far more complicated behavior of the theory near the cosmological singularity than suggested by the standard homogeneous ansätze.