Tensor networks, \(p\)-adic fields, and algebraic curves: arithmetic and the \(\mathrm{AdS}_3 / \mathrm{CFT}_2\) correspondence
type: Item
instance of: scholarly article
title: Tensor networks, \(p\)-adic fields, and algebraic curves: arithmetic and the \(\mathrm{AdS}_3 / \mathrm{CFT}_2\) correspondence
zbMATH Open document ID: 1414.81207
author: Matthew Heydeman
author: Matilde Marcolli
author: Ingmar A. Saberi
author: Bogdan Stoica
published in: Advances in Theoretical and Mathematical Physics
publication date: 2018-10-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/1605.07639
review text: The authors initiate a generalization of the AdS$_3$/CFT$_2$ correspondence in which the bulk geometry is the Bruhat-Tits tree \[ T_p=\mathrm{PGL}(2,\mathbb{Q}_p)/\mathrm{PGL}(2,\mathbb{Z}_p), \] where $\mathbb{Z}_p$ denotes the ring of integers of the $p$-adic field $\mathbb{Q}_p$, and the conformal boundary is the projective line $\mathrm{P}^1(\mathbb{Q}_p)$ whose global conformal symmetry is $\mathrm{SL}(2,\mathbb{Q}_p)$. \par The possibility of a local conformal algebra for $\mathbb{Q}_p$, in analogy with the Virasoro symmetry in two-dimensional conformal field theory is speculative [\textit{E. Melzer}, ``Nonarchimedean conformal field theories,'' Int. J. Mod. Phys. A 4, No. 18, 4877--4908 (1989; \url{doi:10.1142/S0217751X89002065})], so the authors use the global symmetries to constrain the properties of $p$-adic conformal field theories. This is a promising approach since the black hole solutions can be understood in terms of the quotients of the bulk geometry by $p$-adic Schottky groups $\Gamma\subset\mathrm{PGL}(2,\mathbb{Q}_p)$, producing Mumford curves on the boundary. \par The authors study the relation between $T_p$ and tensor networks studied in [\textit{F. Pastawski} et al., J. High Energy Phys. 2015, No. 6, Paper No. 149, 55 p. (2015; Zbl 1388.81094)]. They argue that the natural geometric setting of such a tensor network, for certain uniform tilings, is the Bruhat-Tits tree corresponding to a prime power $q = p^n$, where $n = [\mathbf{k} : \mathbb{Q}_p]/e_\mathbf{k}$, and $e_\mathbf{k}$ is the ramification index of the extension $\mathbf{k}$ of $\mathbb{Q}_p$ of finite degree $[\mathbf{k} : \mathbb{Q}_p] < 1$. \par The authors also study free bosons defined on the tree $T_p$ or possibly a quotient of it by a Schottky group, and on a $p$-adic algebraic curve at the boundary. They show that in the bulk, semi-classical massive scalar fields naturally couple to operators on the conformal boundary in a way that allows for precise holographic reconstructio…
reviewed by: Farhang Loran
Mathematics Subject Classification ID: 81T40
Mathematics Subject Classification ID: 81T20
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 11F85
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 81P70
Mathematics Subject Classification ID: 94B27
Mathematics Subject Classification ID: 81T17
Mathematics Subject Classification ID: 81P40
Mathematics Subject Classification ID: 83C57
zbMATH DE Number: 6945840
zbMATH Keywords: $p$-adic conformal field theory
zbMATH Keywords: AdS/CFT correspondence
MaRDI profile type: Publication
arXiv ID: 1605.07639
DOI: 10.4310/ATMP.2018.V22.N1.A4
description: scientific article