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

TL;DR

This paper provides numerical evidence that low-lying modes around Gaussian solutions in tensor models correspond to metric fluctuations in general relativity, supporting the idea that tensor models can reproduce gravitational dynamics.

Contribution

It offers detailed numerical analysis confirming the agreement between tensor model modes and general relativity, and proposes a renormalization procedure to clarify spectral patterns.

Findings

Low-lying modes match metric fluctuations transverse to coordinate transformations.

Modes follow a massless trajectory with quartic momentum dependence.

Renormalization clarifies spectral patterns and suggests massive trajectories.

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.