Emergent fuzzy geometry and fuzzy physics in $4$ dimensions

TL;DR

This paper uses Monte Carlo simulations to explore emergent fuzzy geometries in matrix models, revealing stable fuzzy spheres in 2D and 4D, and discusses their implications for fuzzy physics and topology change.

Contribution

It demonstrates the emergence of stable fuzzy geometries in bosonic matrix models and explores their stability, universality, and potential for fuzzy field theories.

Findings

Emergent fuzzy spheres ${f S}^2_N$ and ${f S}^2_N\times{f S}^2_N$ observed in simulations.

Stability of fuzzy geometries depends on the dimension and mass parameter $M$.

Potential for topology change and fuzzy sphere as a regulator in quantum field theories.

Abstract

A detailed Monte Carlo calculation of the phase diagram of bosonic IKKT Yang-Mills matrix models in three and six dimensions with quartic mass deformations is given. Background emergent fuzzy geometries in two and four dimensions are observed with a fluctuation given by a noncommutative $U(1)$ gauge theory very weakly coupled to normal scalar fields. The geometry, which is determined dynamically, is given by the fuzzy spheres ${\bf S}^2_N$ and ${\bf S}^2_N\times{\bf S}^2_N$ respectively. The three and six matrix models are in the same universality class with some differences. For example, in two dimensions the geometry is completely stable, whereas in four dimensions the geometry is stable only in the limit $M\longrightarrow \infty$, where $M$ is the mass of the normal fluctuations. The behavior of the eigenvalue distribution in the two theories is also different. We also sketch how we can obtain a stable fuzzy four-sphere ${\bf S}^2_N\times{\bf S}^2_N$ in the large $N$ limit for all values of $M$ as well as models of topology change in which the transition between spheres of different dimensions is observed. The stable fuzzy spheres in two and four dimensions act precisely as regulators which is the original goal of fuzzy geometry and fuzzy physics. Fuzzy physics and fuzzy field theory on these spaces are briefly discussed.