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A Two-Point Hologram for Everything

Tamra Nebabu, Xiao-Liang Qi, Haifeng Tang, Huaijin Wang

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.

A Two-Point Hologram for Everything

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 d boundary and its two-point function data, we derive a concise analytic formula for the dual 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- 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.
Paper Structure (54 sections, 352 equations, 13 figures)

This paper contains 54 sections, 352 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Derive bulk geometry from boundary correlator
  3. Review of Nebabu-Qi's discrete circuit construction
  4. Method I: via unitary matrix integral
  5. A determinant formula for gate angle
  6. Evaluate Toeplitz matrix determinant
  7. Theorem.
  8. Method II: via inverse scattering
  9. Causal kernel decomposition and horizon modes
  10. Inverse scattering
  11. Curvature near horizon and AdS universality
  12. Examples: finite QNMs
  13. Two QNMs
  14. Three QNMs and a concrete de Sitter model
  15. A Gaussian Lindbladian boundary model
  16. ...and 39 more sections

Figures (13)

  • Figure 1: Roadmap from the boundary GFF two-point function to the dual bulk geometry, via two complementary approaches.
  • Figure 2: (a) The Nebabu--Qi circuit Nebabu:2023iox. (b) Causal-wedge reconstruction (cf. \ref{['eq: K is triangular']}). (c) Each "cross" is an SO(2) gate, with the convention in \ref{['eq: definition of gate matrix']}.
  • Figure 3: (a) An illustration of the determinant formula \ref{['eq: general determinant formula from region']}. (b) The parallelepiped used in its derivation.
  • Figure 4: (a) Illustration of Szegő's limit theorem: a Toeplitz matrix may be viewed as a single particle hopping on a one-dimensional lattice with OBC. In the large-$N$ limit, the leading spectrum is well approximated by the corresponding PBC lattice. (b) Illustration of the physical meaning of the three terms in \ref{['eq: free energy expansion']} for the unitary matrix integral. The gray line is the unit circle on complex plane. The blue line and the shaded region represent the classical eigenvalue density $\rho_\text{c}(\theta)$. The wavy lines represent eigenvalue tunneling.
  • Figure 5: (a) Past/future horizon modes defined in \ref{['eq: define past and future mode']}. The horizon is indicated by the green dashed line. (b) The "forward scattering" expansion \ref{['eq: forward scattering, zig-zag']}.
  • ...and 8 more figures