# Quantum Supersymmetric Cosmological Billiards and their Hidden Kac-Moody Structure

## TL;DR

This paper explores the quantum dynamics of a supersymmetric cosmological model, revealing that the reflection operators in the quantum billiard satisfy relations linked to a spinorial extension of a hyperbolic Kac-Moody algebra, indicating deep algebraic structures in quantum cosmology.

## Contribution

It demonstrates that the quantum reflection operators in a supersymmetric cosmological billiard form a spinorial extension of the Weyl group of the AE_3 Kac-Moody algebra, uncovering new algebraic structures in quantum gravity.

## Key findings

- Reflection operators satisfy generalized Coxeter relations.
- Operators form a spinorial extension of AE_3 Weyl group.
- Quantum supersymmetric billiard exhibits hyperbolic Kac-Moody symmetry.

## Abstract

We study the quantum fermionic billiard defined by the dynamics of a quantized supersymmetric squashed three-sphere (Bianchi IX cosmological model within D=4 simple supergravity). The quantization of the homogeneous gravitino field leads to a 64-dimensional fermionic Hilbert space. We focus on the 15- and 20-dimensional subspaces (with fermion numbers N_F=2 and N_F=3) where there exist propagating solutions of the supersymmetry constraints that carry (in the small-wavelength limit) a chaotic spinorial dynamics generalizing the Belinskii-Khalatnikov-Lifshitz classical "oscillatory" dynamics. By exactly solving the supersymmetry constraints near each one of the three dominant potential walls underlying the latter chaotic billiard dynamics, we compute the three operators that describe the corresponding three potential-wall reflections of the spinorial state describing, in supergravity, the quantum evolution of the universe. It is remarkably found that the latter, purely dynamically-defined, reflection operators satisfy generalized Coxeter relations which define a type of spinorial extension of the Weyl group of the rank-3 hyperbolic Kac-Moody algebra AE_3.