Tensor model and dynamical generation of commutative nonassociative fuzzy spaces

TL;DR

This paper demonstrates that a rank-three tensor model can dynamically generate various commutative nonassociative fuzzy spaces, such as fuzzy tori and spheres, with implications for emergent gravity and spacetime structure.

Contribution

It applies the tensor model to generate fuzzy spaces dynamically, showing classical solutions correspond to fuzzy geometries and exploring symmetry breaking related to gravity.

Findings

Fuzzy flat torus and spheres are classical solutions of the tensor model.

Solutions are obtained with the same coupling constants, making cosmological constant and dimensions dynamical.

The model's symmetry is broken at solutions, hinting at gravity as Nambu-Goldstone modes.

Abstract

Rank-three tensor model may be regarded as theory of dynamical fuzzy spaces, because a fuzzy space is defined by a three-index coefficient of the product between functions on it, f_a*f_b=C_ab^cf_c. In this paper, this previous proposal is applied to dynamical generation of commutative nonassociative fuzzy spaces. It is numerically shown that fuzzy flat torus and fuzzy spheres of various dimensions are classical solutions of the rank-three tensor model. Since these solutions are obtained for the same coupling constants of the tensor model, the cosmological constant and the dimensions are not fundamental but can be regarded as dynamical quantities. The symmetry of the model under the general linear transformation can be identified with a fuzzy analog of the general coordinate transformation symmetry in general relativity. This symmetry of the tensor model is broken at the classical solutions. This feature may make the model to be a concrete finite setting for applying the old idea of obtaining gravity as Nambu-Goldstone fields of the spontaneous breaking of the local translational symmetry.