Bending the Bruhat-Tits tree. Part I. Tensor network and emergent Einstein equations

We demonstrate that a unique emergent graph Einstein equation is satisfied by the geometric data encoded in the tensor network, and the graph Einstein tensor automatically recovers the known proposal in the mathematics literature, at least perturbatively order by order in the deformation away from the pure Bruhat-Tits Tree geometry dual to pure CFTs

Lin Chen; Xirong Liu; Ling-Yan Hung

2021

Scholarcy highlights

• Realized the notion of entanglement wedge reconstruction
• We demonstrate that a unique emergent graph Einstein equation is satisfied by the geometric data encoded in the tensor network, and the graph Einstein tensor automatically recovers the known proposal in the mathematics literature, at least perturbatively order by order in the deformation away from the pure Bruhat-Tits Tree geometry dual to pure CFTs
• We study in detail the tensor network proposed in previous research with deformed boundary conditions
• We show that there is virtually a unique way of assigning geometrical data to the tensor network based on its local data so that it satisfies a graph Einstein equation that is self-consistent with the field theory that is known to be encoded in the tensor network
• To summarise our feat slightly differently, the search for Einstein equation is to look for a question whose answer we have — a tensor network carrying the CFT data
• We constructed such a relation guided by the AdS/CFT correspondence, which helped us in finding a bulk action that encodes the same correlation data as the tensor network
• PEPS tensor networks produce CFT partition function by projecting each of the boundary leg to a particular fixed point vector

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