The tensor of the exact circle: Reconstructing geometry

TL;DR

This paper advances tensor model approaches to quantum gravity by demonstrating how to reconstruct the full Riemannian geometry of the circle, including the metric tensor and diffeomorphisms, from algebraic structures.

Contribution

It shows how to reconstruct the complete Riemannian geometry of the circle using tensor models, extending previous topological reconstructions.

Findings

Reconstruction of the metric tensor from tensor data.

Demonstration of diffeomorphism behavior in the tensor formalism.

Explicit example of smoothing point sets via diffeomorphisms.

Abstract

Developing a theory for quantum gravity is one of the big open questions in theoretical high-energy physics. Recently, a tensor model approach has been considered that treats tensors as the generators of commutative non-associative algebras, which might be an appropriate interpretation of the canonical tensor model. In this approach, the non-associative algebra is assumed to be a low-energy description of the so-called associative closure, which gives the full description of spacetime including the high-energy modes. In the previous work it has been shown how to (re)construct a topological space with a measure on it, and one of the prominent examples that was used to develop the framework was the exact circle. In this work we will further investigate this example, and show that it is possible to reconstruct the full Riemannian geometry by reconstructing the metric tensor. Furthermore, it is demonstrated how diffeomorphisms behave in this formalism, firstly by considering a specific class of diffeomorphisms of the circle, namely the ellipses, and subsequently by performing an explicit diffeomorphism to ``smoothen'' sets of points generated by the tensor rank decomposition.