Tensor model and dynamical generation of commutative non-associative fuzzy spaces

Abstract

The rank-three tensor model may be regarded as a theory of dynamical fuzzy spaces, because a fuzzy space is defined by a three-index coefficient of the product between functions on it, fa*fb = Cabcfc. In this paper, this proposal is applied to the dynamical generation of commutative non-associative fuzzy spaces. It is numerically shown that fuzzy flat tori 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 analogue 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 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.