Quantum Supersymmetric Bianchi IX Cosmology

TL;DR

This paper explores the quantum dynamics of a supersymmetric Bianchi IX cosmological model, revealing potential quantum avoidance of singularities and a spectrum of discrete and continuous quantum states influenced by fermionic properties.

Contribution

It introduces a detailed quantum framework for supersymmetric Bianchi IX cosmology, connecting the wavefunction to Kac-Moody algebra structures and analyzing the implications for cosmological singularity avoidance.

Findings

Quantum wavefunction depends on squashing parameters and satisfies Dirac and Klein-Gordon-like equations.

The Hamiltonian relates to a 64-dimensional representation of a Kac-Moody algebra.

Negative squared-mass terms suggest a quantum bounce and a discrete spectrum of states.

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 a Bianchi IX cosmological model. After imposition of the diffeomorphism constraints, the wave function of the Universe becomes a spinor of Spin(8,4) depending on the three squashing parameters, which satisfies Dirac, and Klein-Gordon-like, wave equations describing the propagation of a quantum spinning particle reflecting off spin-dependent potential walls. The algebra of the susy 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 maximally compact sub-algebra of the rank-3 hyperbolic Kac-Moody algebra AE3. The (quartic-in-fermions) squared-mass term entering the Klein-Gordon-like equation has several remarkable properties: 1)it commutes with all the other (Kac-Moody-related) building blocks of the Hamiltonian; 2)it is a quadratic function of the fermion number NF; 3)it is negative in most of the Hilbert space. The latter property leads to a possible quantum avoidance of the singularity ("cosmological bounce"), and suggests imposing the boundary condition that the wavefunction of the Universe vanish when the volume of space tends to zero. The space of solutions is a mixture of "discrete-spectrum states" (explicitely given) and of continuous-spectrum states (parametrized by arbitrary functions entering some initial-value problem). The predominantly negative values of the squared-mass term lead to a "bottle effect" between small and large volume-Universes and to a possible reduction of the continuous spectrum to a discrete spectrum of quantum states looking like excited versions of the Planckian-size Universes described by the discrete states at fermionic levels NF=0 and 1.