An invariant approach to dynamical fuzzy spaces with a three-index variable

Abstract

A dynamical fuzzy space might be described by a three-index variable C_{ab}^c, which determines the algebraic relations f_a f_b =C_{ab}^c f_c among the functions f_a on the fuzzy space. A fuzzy analogue of the general coordinate transformation would be given by the general linear transformation on f_a. I study equations for the three-index variable invariant under the general linear transformation, and show that the solutions can be generally constructed from the invariant tensors of Lie groups. As specific examples, I study SO(3) symmetric solutions, and discuss the construction of a scalar field theory on a fuzzy two-sphere within this framework.

Typos corrected, 12 pages, 8 figures, LaTeX, JHEP class

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• Canonical tensor model through data analysis -- Dimensions, topologies, and geometries --
• The fluctuation spectra around a Gaussian classical solution of a tensor model and the general relativity
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• A canonical rank-three tensor model with a scaling constraint
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• Emergent general relativity in fuzzy spaces from tensor models
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• A renormalization procedure for tensor models and scalar-tensor theories of gravity
• Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models