三角铰链模型中的物质场
Matter fields in triangle-hinge models
摘要：
膜的世界体积理论在数学上等价于三维量子引力耦合到嵌入膜的目标空间坐标所对应的物质场。在最近的一篇论文[arXiv：1503.08812]中，引入了一类新的模型，可以生成三维随机体积，其中每个配置的玻尔兹曼权重由分配给三角形和铰链的值的乘积给出。这些三角铰链模型描述了三维纯引力，其特征在于半单结合代数。在本文中，我们引入物质的自由度的模型，通过染色的方式，他们有局部的相互作用的简单。这是通过简单地扩展原来的三角铰链模型的关联代数，并指定的颜色和相互作用的形式设置的物质场的轮廓。在$D$维时空中，膜的动力学可以通过取颜色集合为$\mathbb{R}^D$来描述。通过采用另一组颜色，我们还可以实现与伊辛模型、$q$态Potts模型或RSOS模型耦合的三维量子引力。实际上，可以将颜色分配给任何维度的单形（四面体，三角形，边和顶点），并且三维着色张量模型可以通过同时着色四面体，三角形和边来实现为三角铰链模型。
英文摘要：
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.
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