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

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.

arXiv comment44 pages, 16 figures, minor corrections

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Cited by in corpus (9)

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• Numerical and analytical analyses of a matrix model with non-pairwise contracted indices
• Phases of a matrix model with non-pairwise index contractions
• Splitting-merging transitions in tensor-vectors systems in exact large-$N$ limits
• Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models
• Phase profile of the wave function of canonical tensor model and emergence of large spacetimes
• Emergence of Lie group symmetric classical spacetimes in canonical tensor model
• Universality of the Wigner-Gurau limit for random tensors
• Counting tensor rank decompositions