Matter fields in triangle-hinge 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.

21 pages, 14 figures. v2: discussions in section 4 improved. v3: title changed, introduction enlarged

References in corpus (5)

• Lost in Translation: Topological Singularities in Group Field Theory
• Physical states in the canonical tensor model from the perspective of random tensor networks
• Random volumes from matrices
• Renormalization procedure for random tensor networks and the canonical tensor model
• An $OSp$ extension of Canonical Tensor Model

Cited by in corpus (7)

• Parallel tempering algorithm for integration over Lefschetz thimbles
• Equation of motion of canonical tensor model and Hamilton-Jacobi equation of general relativity
• Emergent classical geometries on boundaries of randomly connected tensor networks
• An $OSp$ extension of Canonical Tensor Model
• Mother canonical tensor model
• Constraint algebra of general relativity from a formal continuum limit of canonical tensor model
• Triangle-hinge models for unoriented membranes