The lowest modes around Gaussian solutions of tensor models and the general relativity

Abstract

In the previous paper, the number distribution of the low-lying spectra around Gaussian solutions representing various dimensional fuzzy tori of a tensor model was numerically shown to be in accordance with the general relativity on tori. In this paper, I perform more detailed numerical analysis of the properties of the modes for two-dimensional fuzzy tori, and obtain conclusive evidences for the agreement. Under a proposed correspondence between the rank-three tensor in tensor models and the metric tensor in the general relativity, conclusive agreement is obtained between the profiles of the low-lying modes in a tensor model and the metric modes transverse to the general coordinate transformation. Moreover, the low-lying modes are shown to be well on a massless trajectory with quartic momentum dependence in the tensor model. This is in agreement with that the lowest momentum dependence of metric fluctuations in the general relativity will come from the R^2-term, since the R-term is topological in two dimensions. These evidences support the idea that the low-lying low-momentum dynamics around the Gaussian solutions of tensor models is described by the general relativity. I also propose a renormalization procedure for tensor models. A classical application of the procedure makes the patterns of the low-lying spectra drastically clearer, and suggests also the existence of massive trajectories.

31 pages, 8 figures, Added references, minor corrections, a misleading figure replaced

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• 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
• Quantum canonical tensor model and an exact wave function
• Gauge fixing in the tensor model and emergence of local gauge symmetries
• 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