Matter fields in triangle-hinge models

TL;DR

This paper extends triangle-hinge models of 3D quantum gravity by incorporating matter fields through coloring simplices, enabling the modeling of various matter-coupled gravity theories in three dimensions.

Contribution

It introduces a method to add matter degrees of freedom to triangle-hinge models by extending associative algebras, allowing for the simulation of matter-coupled 3D quantum gravity.

Findings

Matter fields are incorporated via coloring simplices with local interactions.

Models can simulate 3D quantum gravity coupled with Ising, Potts, and RSOS models.

Coloring can be applied to simplices of any dimension, enabling colored tensor models.

Abstract

The worldvolume theory of membrane is mathematically equivalent to three-dimensional quantum gravity coupled to matter fields corresponding to the target space coordinates of embedded membrane. In a recent paper [arXiv:1503.08812] a new class of models are introduced that generate three-dimensional random volumes, where the Boltzmann weight of each configuration is given by the product of values assigned to the triangles and the hinges. These triangle-hinge models describe three-dimensional pure gravity and are characterized by semisimple associative algebras. In this paper, we introduce matter degrees of freedom to the models by coloring simplices in a way that they have local interactions. This is achieved simply by extending the associative algebras of the original triangle-hinge models, and the profile of matter field is specified by the set of colors and the form of interactions. The dynamics of a membrane in $D$-dimensional spacetime can then be described by taking the set of colors to be $\mathbb{R}^D$. By taking another set of colors, we can also realize three-dimensional quantum gravity coupled to the Ising model, the $q$-state Potts models or the RSOS models. One can actually assign colors to simplices of any dimensions (tetrahedra, triangles, edges and vertices), and three-dimensional colored tensor models can be realized as triangle-hinge models by coloring tetrahedra, triangles and edges at a time.