Tensor networks, \(p\)-adic fields, and algebraic curves: arithmetic and the \(\mathrm{AdS}_3 / \mathrm{CFT}_2\) correspondence
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 reconstruction. They interpret the radial direction in the tree as a renormalization scale as usual and verify that, quite impressively, the boundary modes contribute to the reconstruction of bulk functions only up to a height determined by their wavelength, while they reconstruct precisely to zero above this height in the tree. \par The authors address the entanglement entropy, which measures the entanglement between the degrees of freedom living on a spatial domain and those living on the complement, in their $p$-adic holography and discuss the subtleties that arise; In the real case the singularities of the entanglement entropy are related to the boundary of the region under consideration, while the $p$-adic numbers have no ordering, and every element of an open set is equally or equally not a boundary element. \par The authors argue that since in their approach to the tensor networks and holographic error-correcting codes, the entanglement entropy of a deleted region is reproduced by counting geodesic lengths in the bulk, the $p$-adic systems that they have considered ``may be closer to tensor network models than their archimedean counterparts.''
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