Spacelike Singularities and Hidden Symmetries of Gravity

Area of Science:

• Theoretical Physics
• High Energy Physics
• Cosmology

Background:

• The BKL-limit describes supergravity dynamics near spacelike singularities.
• Lorentzian Kac-Moody algebras are infinite-dimensional Lie algebras with applications in physics.

Purpose of the Study:

• To explore the connection between supergravity in the BKL-limit and Lorentzian Kac-Moody algebras.
• To reformulate gravitational dynamics using mathematical structures from these algebras.
• To investigate potential symmetries of M-theory and supergravity.

Main Methods:

• Reformulation of gravitational theory in the BKL-limit as billiard motion in hyperbolic space.
• Analysis of dynamics determined by reflections within Lorentzian Coxeter groups.
• Review of mathematical background for Kac-Moody algebras and Coxeter groups.
• Exploration of geodesic sigma models based on Lorentzian Kac-Moody algebras.

Main Results:

• Gravitational dynamics in the BKL-limit are equivalent to billiard dynamics governed by Lorentzian Coxeter groups.
• These Coxeter groups are Weyl groups of infinite-dimensional Kac-Moody algebras, suggesting their role in gravitational symmetries.
• The hyperbolic algebra E10 is presented as a potential underlying symmetry of M-theory.

Conclusions:

• A profound connection exists between supergravity singularities and Lorentzian Kac-Moody algebras.
• This framework offers a new perspective on gravitational dynamics and symmetries.
• The E10 algebra provides a concrete example for exploring M-theory symmetries in cosmological contexts.