It is shown that classical spaces with geometries emerge on boundaries of randomly connected tensor networks with appropriately chosen tensors in the thermodynamic limit. With variation of the tensors, the dimensions of the spaces can be freely chosen, and the geometries, which are curved in general, can be varied. We give the explicit solvable examples of emergent flat tori in arbitrary dimensions, and the correspondence from the tensors to the geometries for general curved cases. The perturbative dynamics in the emergent space is shown to be described by an effective action which is invariant under the spatial diffeomorphism due to the underlying orthogonal group symmetry of the randomly connected tensor network. It is also shown that there are various phase transitions among spaces, including extended and point-like ones, under continuous change of the tensors.
我们研究LLM几何的激发。这些几何形状是由巨引力子凝聚体的反作用产生的。凝聚膜的激发是开弦，这导致了低能下的涌现杨-米尔斯理论。我们研究了这些涌现规范理论的平面极限的动力学，积累了它们是平面${\cal N}=4$ super Yang-Mills的证据。有三点意见支持这一结论：（i）我们证明了原始${\cal N}=4$ super Yang-Mills的平面Hilbert空间与突现规范理论的平面Hilbert空间之间的同构，（ii）我们论证了突现规范理论平面极限的OPE系数为零;我们认为涌现规范理论的反常维数的平面谱是平面${\cal N}=4$超杨-米尔斯的平面谱。尽管突现规范理论的平面极限是平面${\cal N}=4$ super Yang-Mills，但我们解释了为什么突现规范理论不是${\cal N}=4$ super Yang-Mills理论.
Building Bulk Geometry from the Tensor Radon Transform
使用张量Radon变换和相关的数值方法，我们研究了如何在特定的情况下，$\mathrm{AdS}_3/\mathrm{CFT}_2$体几何可以明确地重建从边界纠缠熵。我们发现，给定的边界纠缠熵的$2$ d CFT，这个框架提供了一个定量的措施，检测是否散装双几何微扰（近AdS）限制。在一个定义良好的散装几何存在的情况下，我们明确重建唯一的散装度量张量一旦规范的选择。然后，我们研究了全息和多体系统中静态和动态场景的涌现体几何。除了物理结果，我们的工作表明，数值方法是可行的，有效的研究在AdS/CFT体重构。
