Matter fields in triangle-hinge models

The worldvolume theory of membranes is mathematically equivalent to 3D quantum gravity coupled to matter fields corresponding to the target space coordinates of an embedded membrane. In a recent paper [M. Fukuma et al., J. High Energy Phys. 1507, 088 ( 2015) [arXiv: 1503.08812 [hep-th]]], a new class of models that generate 3D random volumes was introduced, 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 3D 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 such 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 the matter field is specified by the set of colors and the form of the interactions. The dynamics of a membrane in D-dimensional spacetime can then be described by taking the set of colors to be R-D. By taking another set of colors, we can also realize 3D quantum gravity coupled to various spin models such as the Ising model. 3D colored tensor models can also be realized as triangle-hinge models by coloring tetrahedra, triangles, and edges at one time.