  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.
