Bending the Bruhat-Tits Tree II: the p-adic BTZ Black hole and Local Diffeomorphism on the Bruhat-Tits Tree

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, Q p ) 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 padic 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. * Chen and Liu are co-first authors of the manuscript.

Thread/State correspondence: from bit threads to qubit threads

“…This is because essentially F ij can be understood as a generalization of the multipart case of mutual information I. In particular, in the case involving only three elementary regions, we have (23). Furthermore, according to this scheme, one can obtain the expression of PEE in terms of flow fluxes [52]:…”

“…More recently, many enlightening ideas from other fields, such as condensed matter physics, quantum information theory, network flow optimization theory, etc., have entered and benefited the study of the holographic gravity. One of the most striking examples is that inspired by the tensor network method originally used in condensed matter physics as a numerical simulation tool to investigate the wave functions of quantum many-body systems, various holographic tensor network (TN) models have been constructed as toy models of the holographic duality, such as MERA (multiscale entanglement renormalization ansatz) tensor network [15][16][17][18], perfect tensor network [19], random tensor network [20], p-adic tensor network [21][22][23], OSED (one-shot entanglement distillation) tensor network [24][25][26] and so on. For more research on tensor networks in the holographic context, see e.g.…”

Effective field theory on a finite boundary of the Bruhat-Tits tree

“…Relations between T p and tensor network are studied in [16][17][18]. The p-adic version of the Anti-de Sitter/Conformal Field Theory duality [19][20][21] is proposed in [10,22]( p-adic AdS/CFT), which are followed by lots of works, such as [23][24][25][26][27][28][29][30][31][32][33].…”