Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2023
• Main Author: Obster, Dennis
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2023
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/212008
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models. Dennis Obster. SIGMA 19 (2023), 076, 43 pages

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 to find a way to quantise this. In the canonical tensor model, the gravitational degrees of freedom are encoded in a tensorial quantity 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 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 to have a well-defined notion of the topology and geometry for a large class of tensors 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.