Quantum Supersymmetric Cosmology and its Hidden Kac-Moody Structure

TL;DR

This paper explores the quantum behavior of a supersymmetric cosmological model, revealing a hidden Kac-Moody algebra structure that governs the universe's chaotic evolution near singularities.

Contribution

It demonstrates the emergence of a Kac-Moody algebra structure in the quantum dynamics of a supersymmetric cosmological model, linking quantum gravity to infinite-dimensional algebraic symmetries.

Findings

Quantum Hamiltonian constructed from a 64-dimensional fermionic space.

Closure of supersymmetry and Hamiltonian constraint algebra.

Identification of a Kac-Moody algebra structure governing quantum cosmology.

Abstract

We study the quantum dynamics of a supersymmetric squashed three-sphere by dimensionally reducing (to one timelike dimension) the action of D=4 simple supergravity for an SO(3)-homogeneous (Bianchi IX) cosmological model. The quantization of the homogeneous gravitino field leads to a 64-dimensional fermionic Hilbert space. The algebra of the supersymmetry constraints and of the Hamiltonian one is found to close. One finds that the quantum Hamiltonian is built from operators that generate a 64-dimensional representation of the (infinite-dimensional) maximally compact sub-algebra of the rank-3 hyperbolic Kac-Moody algebra AE_3. Some exponentials of these operators generate a spinorial extension of the Weyl group of AE_3 which describe (in the small wavelength limit) the chaotic quantum evolution of the universe near the cosmological singularity.