Lorentzian Fuzzy Spheres

TL;DR

This paper demonstrates that fuzzy spheres are solutions in Lorentzian IKKT matrix models, serving as toy models for noncommutative cosmologies with singularities, and explores their properties and perturbations leading to scalar field theories.

Contribution

It introduces fuzzy sphere solutions in Lorentzian IKKT models and analyzes their geometric and physical properties, including their role as noncommutative cosmological models.

Findings

Fuzzy spheres are solutions in Lorentzian IKKT matrix models.

The commutative limit yields a sphere with non-standard, signature-varying metric.

Perturbations lead to scalar field theories, which can exhibit tachyonic behavior.

Abstract

We show that fuzzy spheres are solutions of ${\it Lorentzian}$ IKKT matrix models. The fuzzy sphere solutions serve as toy models of closed noncommutative cosmologies where big bang/crunch singularities appear only after taking the commutative limit. The commutative limit of these solutions corresponds to a sphere embedded in Minkowski space. This `sphere' has several novel features. The induced metric does not agree with the standard metric on the sphere, and moreover, it does not have a fixed signature. The curvature computed from the induced metric is not constant, has singularities at fixed latitudes (not corresponding to the poles) and is negative. Perturbations are made about the solutions, and are shown to yield a scalar field theory on the sphere in the commutative limit. The scalar field can become tachyonic for a range of the parameters of the theory.