Tensor networks, $p$-adic fields, and algebraic curves: arithmetic and the AdS$_3$/CFT$_2$ correspondence

TL;DR

<3-5 sentence high-level summary> The paper develops a discretized AdS$_3$/CFT$_2$ framework over $p$-adic fields by using the Bruhat--Tits tree $T_p$ as the bulk and the boundary $\mathrm{P^1}(\mathbb{Q}_p)$, preserving the bulk isometries $\mathrm{PGL}(2,\mathbb{Q}_p)$ and unbroken conformal symmetry. It builds holographic tensor-network models from spanning trees of HaPPY tilings and perfect tensors over finite fields, connects bulk reconstruction to boundary $p$-adic CFT data via a Vladimirov-based scalar theory, and extends the picture to higher-genus, Mumford-curve boundary geometries with $p$-adic Schottky groups. The work also discusses $p$-adic entanglement entropy, proposes an adelic viewpoint tying finite- and infinite-place contributions, and outlines future directions including higher-spin generalizations and Drinfeld upper half-plane refinements. This framework provides a mathematically rich, symmetry-preserving avenue to explore holography, tensor networks, and quantum gravity in a non-archimedean setting with potential insights for the archimedean case.

Abstract

One of the many remarkable properties of conformal field theory in two dimensions is its connection to algebraic geometry. Since every compact Riemann surface is a projective algebraic curve, many constructions of interest in physics (which a priori depend on the analytic structure of the spacetime) can be formulated in purely algebraic language. This opens the door to interesting generalizations, obtained by taking another choice of field: for instance, the $p$-adics. We generalize the AdS/CFT correspondence according to this principle; the result is a formulation of holography in which the bulk geometry is discrete---the Bruhat--Tits tree for $\mathrm{PGL}(2,\mathbb{Q}_p)$---but the group of bulk isometries nonetheless agrees with that of boundary conformal transformations and is not broken by discretization. We suggest that this forms the natural geometric setting for tensor networks that have been proposed as models of bulk reconstruction via quantum error correcting codes; in certain cases, geodesics in the Bruhat--Tits tree reproduce those constructed using quantum error correction. Other aspects of holography also hold: Standard holographic results for massive free scalar fields in a fixed background carry over to the tree, whose vertical direction can be interpreted as a renormalization-group scale for modes in the boundary CFT. Higher-genus bulk geometries (the BTZ black hole and its generalizations) can be understood straightforwardly in our setting, and the Ryu-Takayanagi formula for the entanglement entropy appears naturally.

Figures (13)

• Figure 1: The standard representation of the Bruhat--Tits tree. The point at infinity and the center are arbitrary as the tree is homogeneous. Geodesics such as the highlighted one are infinite paths through the tree from $\infty$ to the boundary which uniquely specify elements of $\mathbb{Q}_p$. This path as a series specifies the digits of the decimal expansion of $x \in \mathbb{Q}_2$ in this example. At the $n$th vertex, we choose either $0$ or $1$ corresponding to the value of $x_n$ in the $p^n$th term of $x$. Negative powers of $p$ correspond to larger $p$-adic norms as we move towards the point $\infty$.
• Figure 2: An alternative representation of the Bruhat--Tits Tree (for $p=3$) in which we have unfolded the tree along the $0$ geodesic. The action of elements of $\mathop{\mathrm{PGL}}\nolimits(2, \mathbb{Q}_p)$ acts by translating the entire tree along different possible geodesics. In this example we translate along the $0$ geodesic, which can be thought of as multiplication of each term in a $p$-adic decimal expansion by $p$. This map has two fixed points at $0$ and $\infty$. In this "unfolded" form, a point in $\mathop{\mathrm{P^1}}\nolimits(\mathbb{Q}_p)$ is specified by a geodesic that runs from $\infty$ and follows the $0$ geodesic until some level in the tree where it leaves the $0$ geodesic towards the boundary. The $p$-adic norm is simply $p$ to the inverse power of the point where it leaves the $0$ geodesic (so leaving "sooner" leads to a larger norm, and later to a smaller norm).
• Figure 3: Obtaining trees for ramified and unramified quadratic extensions from the Bruhat--Tits tree of $\mathbb{Q}_p$.
• Figure 4: Fundamental domain and quotient for the Euclidean BTZ black hole. Compare with the $p$-adic BTZ geometry, shown in Fig. \ref{['BTZpadicFig']}.
• Figure 5: Fundamental domains for the action of $\Gamma$ on $\mathbf{H}^3$.

Theorems & Definitions (2)

• Proposition 1
• proof