M-theory and E10: Billiards, Branes, and Imaginary Roots

TL;DR

This paper explores the connection between E10 Lie algebra roots and physical objects in M-theory, proposing that real roots relate to fluxes and instantons, while imaginary roots correspond to branes and particles, enriching the understanding of billiard cosmology.

Contribution

It introduces a novel interpretation of E10 roots, including imaginary roots, as physical branes and particles in M-theory, expanding the algebraic framework of billiard cosmology.

Findings

Real roots correspond to fluxes and instantons.

Imaginary roots are related to branes and particles.

Interactions are classified by root inner products.

Abstract

Eleven dimensional supergravity compactified on $T^{10}$ admits classical solutions describing what is known as billiard cosmology - a dynamics expressible as an abstract (billiard) ball moving in the 10-dimensional root space of the infinite dimensional Lie algebra E10, occasionally bouncing off walls in that space. Unlike finite dimensional Lie algebras, E10 has negative and zero norm roots, in addition to the positive norm roots. The walls above are related to physical fluxes that, in turn, are related to positive norm roots (called real roots) of E10. We propose that zero and negative norm roots, called imaginary roots, are related to physical branes. Adding `matter' to the billiard cosmology corresponds to adding potential terms associated to imaginary roots. The, as yet, mysterious relation between E10 and M-theory on $T^{10}$ can now be expanded as follows: real roots correspond to fluxes or instantons, and imaginary roots correspond to particles and branes (in the cases we checked). Interactions between fluxes and branes and between branes and branes are classified according to the inner product of the corresponding roots (again in the cases we checked). We conclude with a discussion of an effective Hamiltonian description that captures some features of M-theory on $T^{10}.$