Abstract
As an extended companion paper to [1], we elaborate in detail how the tensor network construction of a p-adic CFT encodes geometric information of a dual geometry even as we deform the CFT away from the fixed point by finding a way to assign distances to the tensor network. In fact we demonstrate that a unique (up to normalizations) 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. Once the dust settles, it becomes apparent that the assigned distance indeed corresponds to some Fisher metric between quantum states encoding expectation values of bulk fields in one higher dimension. This is perhaps a first quantitative demonstration that a concrete Einstein equation can be extracted directly from the tensor network, albeit in the simplified setting of the p-adic AdS/CFT.
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 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
Summary
The asymptotic boundary of the Bruhat-Tits tree is the Qp line. The tensor network has to be cutoff near the asymptotic boundary of the Bruhat-Tits tree — analogous to the cutoff introduced in AdS space. One can see that the computation of CFT correlation functions naturally reduces to sums of Witten diagrams, constructed from bulk-boundary propagators Ga(xi, v) meeting at bulk vertices v. A three point vertex is weighted by Cabc These suggest that at least where the operator insertion at the boundary is sparse, the tensor network can be described by an emergent bulk quantum field theory with action given by. That the two point function read off from the insertion of extra bulk legs is given by φaxφay = p−∆ad(x,y) In this normalization, the action is expressed as ζp (2∆a) 2p∆a xy x ζp The factor 1/(p + 1) is to cancel the p + 1 times overcounting since each vertex attaches to p + 1 edges
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