Emergent Einstein Equation in p-adic CFT Tensor Networks

TL;DR

This paper demonstrates that a tensor network for p-adic CFTs on a deformed Bruhat-Tits tree satisfies an emergent Einstein equation, linking tensor network geometry with gravitational dynamics in a novel p-adic setting.

Contribution

It introduces an emergent graph Einstein equation in p-adic tensor networks, connecting boundary CFT data with bulk geometric dynamics in a new mathematical framework.

Findings

The geometry satisfies a unique emergent Einstein equation.

The graph curvature naturally arises from the Einstein equation's consistency.

Perturbative analysis confirms the correspondence with bulk matter actions.

Abstract

We take the tensor network describing explicit p-adic CFT partition functions proposed in [1], and considered boundary conditions of the network describing a deformed Bruhat-Tits (BT) tree geometry. We demonstrate that this geometry satisfies an emergent graph Einstein equation in a unique way that is consistent with the bulk effective matter action encoding the same correlation function as the tensor network, at least in the perturbative limit order by order away from the pure BT tree. Moreover, the (perturbative) definition of the graph curvature in the Mathematics literature naturally emerges from the consistency requirements of the emergent Einstein equation. This could provide new insights into the understanding of gravitational dynamics potentially encoded in more general tensor networks.