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Boundary Kinematic Space

Andreas Karch, James Sully, Christoph F. Uhlemann, Devin G. E. Walker

TL;DR

The paper extends the kinematic-space program to boundary CFTs by defining a boundary kinematic space for the boundary OPE (bOPE) as a subspace of the ambient kinematic space and establishing a holographic dictionary where bOPE blocks map to geodesic operators. It shows that the bOPE data, via a weighted X-ray transform and a spectrum of radial bulk modes, suffices to reconstruct the bulk geometry by solving an inverse Sturm-Liouville problem, thus illustrating how kinematic space can illuminate bulk dynamics beyond pure kinematics. The framework relies on AdS$_d$ slicing holography, mirror/defect constructions, and defect Casimir equations, providing a pathway to recover warp factors and bulk fields from boundary data with boundary symmetry guiding the reconstruction. The work also highlights avenues for future research, including background-independent operator mappings, consistency relations across multiple kinematic spaces, and the role of long geodesics in more intricate holographic setups.

Abstract

We extend kinematic space to a simple scenario where the state is not fixed by conformal invariance: the vacuum of a conformal field theory with a boundary (bCFT). We identify the kinematic space associated with the boundary operator product expansion (bOPE) as a subspace of the full kinematic space. In addition, we establish representations of the corresponding bOPE blocks in a dual gravitational description. We show how the new kinematic dictionary and the dynamical data in bOPE allows one to reconstruct the bulk geometry. This is evidence that kinematic space may be a useful construction for understanding bulk physics beyond just kinematics.

Boundary Kinematic Space

TL;DR

The paper extends the kinematic-space program to boundary CFTs by defining a boundary kinematic space for the boundary OPE (bOPE) as a subspace of the ambient kinematic space and establishing a holographic dictionary where bOPE blocks map to geodesic operators. It shows that the bOPE data, via a weighted X-ray transform and a spectrum of radial bulk modes, suffices to reconstruct the bulk geometry by solving an inverse Sturm-Liouville problem, thus illustrating how kinematic space can illuminate bulk dynamics beyond pure kinematics. The framework relies on AdS slicing holography, mirror/defect constructions, and defect Casimir equations, providing a pathway to recover warp factors and bulk fields from boundary data with boundary symmetry guiding the reconstruction. The work also highlights avenues for future research, including background-independent operator mappings, consistency relations across multiple kinematic spaces, and the role of long geodesics in more intricate holographic setups.

Abstract

We extend kinematic space to a simple scenario where the state is not fixed by conformal invariance: the vacuum of a conformal field theory with a boundary (bCFT). We identify the kinematic space associated with the boundary operator product expansion (bOPE) as a subspace of the full kinematic space. In addition, we establish representations of the corresponding bOPE blocks in a dual gravitational description. We show how the new kinematic dictionary and the dynamical data in bOPE allows one to reconstruct the bulk geometry. This is evidence that kinematic space may be a useful construction for understanding bulk physics beyond just kinematics.

Paper Structure

This paper contains 24 sections, 79 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The kinematic dictionary
  3. A kinematic space for OPE blocks
  4. A kinematic space for geodesic operators
  5. The kinematic dictionary
  6. Operator products in bCFT
  7. bOPE
  8. Kinematic space for bCFT blocks
  9. Boundary kinematic space as a subspace of Kg
  10. A kinematic dictionary for bCFT
  11. Holographic duals for bCFTs
  12. Geodesic operators for bCFT
  13. Geodesics
  14. bOPE geodesic operators
  15. Boundary Conditions
  16. ...and 9 more sections

Figures (2)

  • Figure 1: Two-point function crossing symmetry underlying the conformal boundary bootstrap. In the presence of boundaries, one-point functions need not vanish, so the sum on the right hand side is non-trivial. Also, two-point functions can depend on one non-trivial conformally invariant cross-ratio and so there is already interesting information contained at that level.
  • Figure 2: Folding of a geodesic across the defect in a dCFT to a long geodesic in a bCFT. $C$ denotes the conformal boundary of the asymptotically-AdS geometry while $D$/$B$ labels the extension of the defect/boundary into the bulk.