论文标题
关于$ 2+1d $重力的$ su_q（2）$的动态出现，带有宇宙常数
On the dynamical emergence of $SU_q(2)$ from the regularization of $2+1D$ gravity with cosmological constant
论文作者
论文摘要
已经证明，爱因斯坦 - 希尔伯特重力作用的缩小相位的量化$ 2+1d $已被证明在量子级别的拓扑量子场理论中带来了$ su_q（2）$量子组对称结构的出现。我们在此解决了同样的问题，但是从运动学理论的Kinematical $ su（2）$（2）$（量子）$ 2+1d $重力，在Palatini形式主义中具有非零宇宙常数，随后施加了约束。因此，我们显示了$ su_q（2）$ Quantum grout在自旋泡沫框架内的量子级别的动态出现。正则化曲率约束负责$ SU_Q（2）$的有效表示，这些表示的任何Wilson Loop在$ SU（2）$组元素上评估的任何Wilson Loop均可编码由宇宙学常数引起的时空曲率离散化。向自旋网络基础的扩展，以及其通用状态之间的任何过渡幅度的扩展，使我们能够完整地得出$ su_q（2）$的重新耦合理论。我们为两个循环状态和编码三价顶点的旋转网络的标量产品提供了建设性示例。我们进一步评论曲率约束的实现产生的差异对称性，并最终明确得出了Turaev-Viro模型的分区函数振幅。
The quantization of the reduced phase-space of the Einstein-Hilbert action for gravity in $2+1D$ has been shown to bring about the emergence, at the quantum level, of a topological quantum field theory endowed with an $SU_q(2)$ quantum group symmetry structure. We hereby tackle the same problem, but start from the kinematical $SU(2)$ (quantum) Hilbert space of the theory of $2+1D$ gravity with non-zero cosmological constant in the Palatini formalism, and subsequently impose the constraints. We hence show the dynamical emergence of the $SU_q(2)$ quantum group at the quantum level within the spin-foam framework. The regularized curvature constraint is responsible for the effective representations of $SU_q(2)$ that are recovered for any Wilson loop evaluated at the $SU(2)$ group element that encodes the discretization of the space-time curvature induced by the cosmological constant. The extension to the spin-network basis, and consequently to any transition amplitude between its generic states, enables us to derive in full generality the recoupling theory of $SU_q(2)$. We provide constructive examples for the scalar product of two loop states and spin-networks encoding trivalent vertices. We further comment on the diffeomorphism symmetry generated by the implementation of the curvature constraint, and finally derive explicitly the partition function amplitude of the Turaev-Viro model.
