A Two-Point Hologram for Everything

TL;DR

This paper develops a holographic dictionary that does not rely on symmetry or asymptotic AdS conditions, deriving bulk geometries from boundary two-point functions and exploring their properties in various models.

Contribution

It introduces a novel method to derive bulk geometries from boundary data without symmetry assumptions, expanding holographic duality beyond traditional frameworks.

Findings

Derived an explicit formula for bulk geometry from boundary two-point functions.

Identified conditions for positive and negative curvature in the dual geometry.

Analyzed the large-q SYK model's temperature-dependent near-horizon curvature.

Abstract

Known holographic dictionaries, especially AdS/CFT, rely on symmetry matching between the bulk and the boundary. We take a step toward a holographic dictionary with no symmetry requirement and without assuming the geometry being asymptotically AdS. Starting from any interacting Majorana generalized free field on a $(0+1)$d boundary and its two-point function data, we derive a concise analytic formula for the dual $(1+1)$d bulk geometry, borrowing techniques from unitary matrix integral and inverse scattering. Using this formula, we compute the near-horizon curvature, give conditions for positive versus negative curvature, and identify simple boundary models with de Sitter or anti-de Sitter near-horizon duals. We also study the large-$q$ SYK model, finding an unusual temperature dependence of the near-horizon curvature, related to the discrepancy between physical temperature and the ``fake disk'' temperature. We also construct, directly from boundary operators, approximate algebras generated by null translations and boost that become exact at the bifurcate horizon.