Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models

TL;DR

This paper explores how tensor models can be interpreted as encoding spacetime geometry, establishing a link between algebraic tensor structures and the topology and measure of Riemannian manifolds.

Contribution

It introduces a method to recover spacetime topology and geometry from tensorial quantities in canonical tensor models, advancing understanding of their physical interpretation.

Findings

Recovered topology and measure of compact Riemannian manifolds from tensors

Generalized topology and geometry for a broad class of tensors using rank decomposition

Demonstrated emergence of spacetime structures from algebraic tensor models

Abstract

The quest for a consistent theory for quantum gravity is one of the most challenging problems in theoretical high-energy physics. An often-used approach is to describe the gravitational degrees of freedom by the metric tensor or related variables, and finding a way to quantise this. In the canonical tensor model, the gravitational degrees of freedom are encoded in a tensorial quantity $P_{abc}$, and this quantity is subsequently quantised. This makes the quantisation much more straightforward mathematically, but the interpretation of this tensor as a spacetime is less evident. In this work we take a first step towards fully understanding the relationship to spacetime. By considering $P_{abc}$ as the generator of an algebra of functions, we first describe how we can recover the topology and the measure of a compact Riemannian manifold. Using the tensor rank decomposition, we then generalise this principle in order to have a well-defined notion of the topology and geometry for a large class of tensors $P_{abc}$. We provide some examples of the emergence of a topology and measure of both exact and perturbed Riemannian manifolds, and of a purely algebraically-defined space called the semi-local circle.