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

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.

Bending the Bruhat-Tits tree. Part II. The p-adic BTZ black hole and local diffeomorphism on the Bruhat-Tits tree

“…The form of the flat connection ended up looking completely parallel to the SL(2, C) flat connection that follows from the Euclidean AdS 3 metric. 1 Moreover the Wilson line junctions evaluated in this flat connection coincides with the tensor network that we constructed to recover the p-adic partition function.…”

“…In our prequel [1], the tensor network reconstruction of the p-adic CFT partition function is shown to naturally recover a bulk action and a covariantized matter action. The edge lengths did not show up as an independent set of operators in the CFT, but rather as a Fisher information metric between states with various non-local operator insertion.…”

“…Here, we will compute the CFT correlation functions using the tensor network we have proposed in [16]. A very brief review of p-adic CFT and our tensor network can be found also in the first of this series [1,20].…”

“…One can also define bulk operator φ a inserted at a bulk vertex v. This is achieved by fusing a label a to the fusion tree at v defined by the OPE coefficients C abc . For further details one can refer to volume one of our current series [1].…”

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In this sequel to [1], we take up a second approach in bending the Bruhat-Tits tree. Inspired by the BTZ black hole connection, we demonstrate that one can transplant it to the Bruhat-Tits tree, at the cost of defining a novel "exponential function" on the p-adic numbers that is hinted by the BT tree.

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