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A dynamical fuzzy space might be described by a three-index variable , which determines the algebraic relations 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. We 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, we study SO(3) symmetric solutions and discuss the construction of a scalar field theory on a fuzzy two-sphere within this framework.

Tensor models can be interpreted as theory of dynamical fuzzy spaces. In this paper, I study numerically the fluctuation spectra around a Gaussian classical solution of a tensor model, which represents a fuzzy flat space in arbitrary dimensions. It is found that the momentum distribution of the low-lying low-momentum spectra is in agreement with that of the metric tensor modulo the general coordinate transformation in the general relativity at least in the dimensions studied numerically, i.e. one to four dimensions. This result suggests that the effective field theory around the solution is described in a similar manner as the general relativity.

In the paper arXiv:0706.1618[hep-th], 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-3 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 R²-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.

Tensor models are more-index generalizations of the so-called matrix models, and provide models of quantum gravity with the idea that spaces and general relativity are emergent phenomena. In this paper, a renormalization procedure for the tensor models whose dynamical variable is a totally symmetric real three-tensor is discussed. It is proven that configurations with certain Gaussian forms are the attractors of the three-tensor under the renormalization procedure. Since these Gaussian configurations are parametrized by a scalar and a symmetric two-tensor, it is argued that, in general situations, the infrared dynamics of the tensor models should be described by scalar-tensor theories of gravity.

Tensor models are generalization of matrix models, and are studied as models of quantum gravity. It is shown that the symmetry of the rank-three tensor models is generated by a hierarchy of n-ary algebras starting from the usual commutator, and the 3-ary algebra symmetry reported in the previous paper is just a single sector of the whole structure. The condition for the Leibnitz rules of the n-ary algebras is discussed from the perspective of the invariance of the underlying algebra under the n-ary transformations. It is shown that the n-ary transformations which keep the underlying algebraic structure invariant form closed finite n-ary Lie subalgebras. It is also shown that, in physical settings, the 3-ary transformation practically generates only local infinitesimal symmetry transformations, and the other more nonlocal infinitesimal symmetry transformations of the tensor models are generated by higher n-ary transformations.

By extending the algebraic description of the bosonic rank-three tensor models, a general framework for super rank-three tensor models and correspondence to super fuzzy spaces is proposed. The corresponding super fuzzy spaces must satisfy a certain cyclicity condition on the algebras of functions on them. Due to the cyclicity condition, the symmetry of the super rank-three tensor models are represented by super n-ary transformations. The Leibnitz rules and the fundamental identities for the super n-ary transformations are discussed from the perspective of the symmetry of the algebra of a fuzzy space. It is shown that the super n-ary transformations of finite orders which conserve the algebra of a fuzzy space form a finite closed n-ary super Lie algebra. Super rank-three tensor models would be of physical interest as background independent models for dynamical generation of supersymmetric fuzzy spaces, in which quantum corrections are under control.

Canonical formalism of the rank-three tensor model has recently been proposed, in which "local" time is consistently incorporated by a set of first class constraints. By brute-force analysis, this paper shows that there exist only two forms of a Hamiltonian constraint which satisfies the following assumptions: (i) A Hamiltonian constraint has one index. (ii) The kinematical symmetry is given by an orthogonal group. (iii) A consistent first class constraint algebra is formed by a Hamiltonian constraint and the generators of the kinematical symmetry. (iv) A Hamiltonian constraint is invariant under time reversal transformation. (v) A Hamiltonian constraint is an at most cubic polynomial function of canonical variables. (vi) There are no disconnected terms in a constraint algebra. The two forms are the same except for a slight difference in index contractions. The Hamiltonian constraint which was obtained in the previous paper and behaved oddly under time reversal symmetry can actually be transformed to one of them by a canonical change of variables.

In this paper, a short review of the operator/Feynman diagram/dessin d’enfants correspondence in the rank 3 tensor model is presented, and the cut and join operation is given in the language of dessin d’enfants as a straightforward development. We classify operators of the rank 3 tensor model up to level 5 with dessin d’enfants. (Based on the talk given by R. Y. at the international workshop “Randomness, Integrability and Representation Theory in Quantum Field Theory” at the Osaka City University Media Center on March 25, 2021.)

In this paper, to understand space–time dynamics in the canonical tensor model of quantum gravity for the positive cosmological constant case, we analytically and numerically study the phase profile of its exact wave function in a coordinate representation, instead of the momentum representation analyzed so far. A saddle point analysis shows that Lie group symmetric space–times are strongly favored due to abundance of continuously existing saddle points, giving an emergent fluid picture. The phase profile suggests that spatial sizes grow in “time,” where sizes are measured by the tensor-geometry correspondence previously introduced using tensor rank decomposition. Monte Carlo simulations are also performed for a few small $N$ cases by applying a re-weighting procedure to an oscillatory integral which expresses the wave function. The results agree well with the saddle point analysis, but the phase profile is subject to disturbances in a large space–time region, suggesting existence of light modes there and motivating future computations of primordial fluctuations from the perspective of canonical tensor model.

To enhance the accuracy results of the change detection of buildings utilizing high-resolution remote sensing (HRRS) images, a novel method was proposed by combining both tensor and deep belief network (DBN). To optimize the description of the essential characteristics for changes in buildings, a tensor-based structure covering time-space-spectrum-shadow features integrated into the model (TSSS-Cube) is proposed. The changes occurring as a combination of shadow and spectral features and spatio-temporal autocorrelation at each pixel are represented by a third-order tensor to maintain the structural information and the constraint integrity between them. Then, a restricted Boltzmann machine (TC-RBM) that can be directly used to process TSSS-Cube data is designed, and the support tensor machine (STM) is used to replace the conventional backpropagation neural network at the top of the DBN to construct a multi-tensor deep belief network (MTR-DBN) composing of multi-layer TC-RBMs and an STM classifier. Finally, the multi-layer TC-RBMs in MTR-DBN are trained layer by layer, and the global parameters of the MTR-DBN are optimized by combining a limited number of labeled data and fine-tuning the STM classifier. The implementation of both supervised and unsupervised learning methods comprehensively provides advantages to increase the accuracy result of the MTR-DBN network to detect changes. Three representative different sub-regions are selected from the whole original experimental area respectively for building change detection experiments, and a dataset composed of double-temporal HRRS images in 2012 and 2016 is used as the related experimental dataset. The experimental results show that both a change detection accuracy result with a higher average and better detection efficiency is attained by the proposed method called the MTR-DBN when compared with other similar methods.

A canonical formalism of the rank-three tensor model with the notion of local time is proposed. The consistency of the local time evolution is guaranteed by imposing that local Hamiltonians and the so(N) kinematical symmetry of the tensor model should form a first class constraint algebra. By imposing some physically reasonable assumptions, it is shown that there exist only two such local Hamiltonians with a slight difference in index contraction. The first class constraint algebra is shown to approach the DeWitt constraint algebra of the general relativity in a certain locality limit. Quantization of the system is briefly discussed.