Abstract
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. We demonstrate that the PGL(2, Qp) Wilson lines [2] evaluated on this analogue BTZ connection is indeed consistent with correlation functions of a CFT at finite temperatures. We demonstrate that these results match up with the tensor network reconstruction of the p-adic AdS/CFT with a different cutoff surface at the asymptotic boundary, and give explicit coordinate transformations that relate the analogue p-adic BTZ background and the “pure” Bruhat-Tits tree background. This is an interesting demonstration that despite the purported lack of descendents in p-adic CFTs, there exists non-trivial local Weyl transformations in the CFT corresponding to diffeomorphism in the Bruhat-Tits tree.
Highlights
Ask whether one can include dynamics of the background so that the bulk theory becomes an analogue of gravitational theory on AdS
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
We demonstrate that the PGL(2, Qp) Wilson lines [2] evaluated on this analogue BTZ connection is consistent with correlation functions of a CFT at finite temperatures
Summary
We will describe a novel coordinates system that is most natural for the BT tree when it is arranged to describe the analogue BTZ black hole. We will describe how this black hole coordinates can be related to the usual Poincare coordinates via a coordinate transformation analogous to the corresponding transformation in AdS3 [17]
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