Tensor model and dynamical generation of commutative nonassociative fuzzy spaces

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.

Adding discussions on effective geometry, a note added, four references added, other minor changes, 27 pages, 17 figures

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Cited by in corpus (17)

• Canonical tensor models with local time
• Uniqueness of canonical tensor model with local time
• Super tensor models, super fuzzy spaces and super n-ary transformations
• Tensor models and hierarchy of n-ary algebras
• A toy model of open membrane field theory in constant 3-form flux
• Gauge fixing in the tensor model and emergence of local gauge symmetries
• The fluctuation spectra around a Gaussian classical solution of a tensor model and the general relativity
• The lowest modes around Gaussian solutions of tensor models and the general relativity
• Canonical tensor model through data analysis -- Dimensions, topologies, and geometries --
• A canonical rank-three tensor model with a scaling constraint
• $SO(3)$-invariant phase of the $O(N{)}^3$ tensor model
• Tensor models and 3-ary algebras
• Numerical and analytical analyses of a matrix model with non-pairwise contracted indices
• A renormalization procedure for tensor models and scalar-tensor theories of gravity
• Emergent general relativity in the tensor models possessing Gaussian classical solutions
• Fuzzy spaces from tensor models, cyclicity condition, and n-ary algebras
• Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models