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

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 $T^{10}.$

68pp, references added as well as minor corrections

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Cited by in corpus (25)

• Spacelike Singularities and Hidden Symmetries of Gravity
• Extended E8 Invariance of 11-Dimensional Supergravity
• Higher order M theory corrections and the Kac-Moody algebra E10
• Counting supersymmetric branes
• An E9 multiplet of BPS states
• IIB supergravity and E10
• Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits
• FRW solutions and holography from uplifted AdS/CFT
• Integrability of Supergravity Billiards and the generalized Toda lattice equation
• E10 Cosmology
• Exotic E11 branes as composite gravitational solutions
• Cosmic Billiards with Painted Walls in Non-Maximal Supergravities: a worked out example
• The general pattern of Kac Moody extensions in supergravity and the issue of cosmic billiards
• Hyperbolic Weyl groups and the four normed division algebras
• Eisenstein series for infinite-dimensional U-duality groups
• Fermionic Kac-Moody Billiards and Supergravity
• The monodromy of T-folds and T-fects
• Supergravity Black Holes and Billiards and Liouville integrable structure of dual Borel algebras
• Five-dimensional Supergravity and Hyperbolic Kac-Moody Algebra G2H
• E10 Orbifolds
• Time-like T-duality algebra
• Some Algebraic Aspects of Half-BPS Bound States in M-Theory
• Sugawara-type constraints in hyperbolic coset models
• Complexity and the Big Bang
• N-Extended Lorentzian Kac-Moody algebras