曲面四面体的面整体编码：群值矩映射的相空间
Encoding Curved Tetrahedra in Face Holonomies: a Phase Space of Shapes from Group-Valued Moment Maps
摘要：
我们提出了一个推广的Minkowski的经典定理重建四面体从代数数据均匀弯曲的空间。欧几里得的概念，如法向量的脸被取代的列维-奇维塔holonomies周围的每个四面体的脸。这允许在一个统一的框架内重建球面和双曲四面体。为了覆盖代数数据中编码的所有扇区，引入了一种新的双曲单形。推广相空间的形状相关联的平面四面体导致组值矩映射和准泊松空间。这些离散的几何提供了一个自然的舞台，考虑量子化的重力，包括一个宇宙学常数。与圈量子引力的自旋网络态的关系提供了这一点的具体实现。因此，这项工作提供了一个自下而上的理由出现的变形规范对称性和量子群的3+1维协变圈量子引力在存在一个宇宙学常数。
英文摘要：
We present a generalization of Minkowski's classic theorem on the reconstruction of tetrahedra from algebraic data to homogeneously curved spaces. Euclidean notions such as the normal vector to a face are replaced by Levi-Civita holonomies around each of the tetrahedron's faces. This allows the reconstruction of both spherical and hyperbolic tetrahedra within a unified framework. A new type of hyperbolic simplex is introduced in order for all the sectors encoded in the algebraic data to be covered. Generalizing the phase space of shapes associated to flat tetrahedra leads to group valued moment maps and quasi-Poisson spaces. These discrete geometries provide a natural arena for considering the quantization of gravity including a cosmological constant. A concrete realization of this is provided by the relation with the spin-network states of loop quantum gravity. This work therefore provides a bottom-up justification for the emergence of deformed gauge symmetries and quantum groups in 3+1 dimensional covariant loop quantum gravity in the presence of a cosmological constant.
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