Kaluza-Klein ansatz from Lorentzian quantum gravity on the fuzzy sphere

TL;DR

This paper explores how quantum gravity effects on a fuzzy sphere fibre can lead to the emergence of gravity and Yang-Mills theories, supporting Kaluza-Klein ideas through noncommutative geometry.

Contribution

It demonstrates that quantum gravity on a fuzzy sphere naturally enforces a spherical fibre geometry consistent with Kaluza-Klein models.

Findings

The expected metric value is spherical.

The fibre size can be considered constant due to renormalisation.

Gravity plus Yang-Mills emerge from quantum gravity effects.

Abstract

If Kaluza-Klein ideas were correct as an explanation of Yang-Mills and General Relativity on spacetime, the extra fibre geometry would have to be a sphere of constant size of the order of 10 Planck lengths, hence subject to quantum gravity corrections. Conversely, it was shown in previous work that modelling such corrections by noncommutative coordinates indeed forces the Kaluza-Klein cylinder ansatz form of the metric, and we now propose that the remaining restrictions needed come from quantum gravity on the fibre. Working with a fuzzy sphere fibre, we find that the expected value of the metric is indeed spherical and we propose that it can be taken as of constant size due to freedom in the renormalisation of divergences. In this way, we outline a mechanism whereby the observed structure of gravity plus Yang-Mills can emerge at low energies as a consequence of quantum gravity effects.