Geodesic string condensation from symmetric tensor gauge theory: A unifying framework of holographic toy models

We propose that there is a universal picture for different constructions of holographic toy models, and its continuous limit can be described by a gravitylike field theory, namely, the rank-2 U(1) gauge theory. First, we show that two different toy models for holography---the perfect tensor networks and the hyperbolic fracton models---are both equivalent to a picture of evenly distributed bit threads on geodesics in the anti--de Sitter space. We name this picture ``geodesic string condensation.'' It is actually a natural leading-order approximation to the holographic entanglement structure. Then, we reason that the rank-2 U(1) gauge theory with linearized diffeomorphism as its gauge symmetry, also known as a case of Lifshitz gravity, is the bulk field theory that gives rise to this picture. The Gauss's laws and spatial curvature require the geodesic gauge field lines, instead of the local loops (magnetic fields), to be the fundamental dynamical variables, which lead to geodesic string condensation. These results provide an intuitive way to understand the entanglement structure of gravity in anti--de Sitter/conformal field theory.