Canonical tensor model through data analysis -- Dimensions, topologies, and geometries --

TL;DR

This paper introduces a data analysis approach using tensor decomposition and persistent homology to extract topological and geometric information from the canonical tensor model, linking tensor dynamics to spacetime structures.

Contribution

It presents a novel method to interpret the CTM's tensors as spacetime topologies and geometries using established data analysis techniques, demonstrating its general applicability across dimensions.

Findings

Agreement with general relativity in fuzzy sphere models

Method applicable to any dimensions and topologies

Initial results support the CTM as a spacetime model

Abstract

The canonical tensor model (CTM) is a tensor model in Hamilton formalism and is studied as a model for gravity in both classical and quantum frameworks. Its dynamical variables are a canonical conjugate pair of real symmetric three-index tensors, and a question in this model was how to extract spacetime pictures from the tensors. We give such an extraction procedure by using two techniques widely known in data analysis. One is the tensor-rank (or CP, etc.) decomposition, which is a certain generalization of the singular value decomposition of a matrix and decomposes a tensor into a number of vectors. By regarding the vectors as points forming a space, topological properties can be extracted by using the other data analysis technique called persistent homology, and geometries by virtual diffusion processes over points. Thus, time evolutions of the tensors in the CTM can be interpreted as topological and geometric evolutions of spaces. We have performed some initial investigations of the classical equation of motion of the CTM in terms of these techniques for a homogeneous fuzzy circle and homogeneous two- and three-dimensional fuzzy spheres as spaces, and have obtained agreement with the general relativistic system obtained previously in a formal continuum limit of the CTM. It is also demonstrated by some concrete examples that the procedure is general for any dimensions and topologies, showing the generality of the CTM.