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Unimodular time in JT gravity: a holographic clock

Altay Etkin, Farbod-Sayyed Rassouli

Abstract

How is a ''bulk clock'' encoded holographically? We address this in Jackiw-Teitelboim (JT) gravity, where a natural physical clock emerges by promoting the vacuum energy to a dynamical variable: the vacuum cosmological constant becomes a top form degree of freedom conjugate to spacetime volume, thereby defining a notion of bulk physical time. This construction is naturally formulated in the Henneaux-Teitelboim (HT) framework. We show that the boundary dynamics is the Schwarzian mode coupled to a free particle on $U(1)$, matching the universal low-energy effective action of the complex SYK model. By further clarifying the role of the vacuum cosmological constant as a top form, we establish the equivalence between JT gravity coupled to two-dimensional Maxwell theory and 2d HT gravity via an explicit field redefinition. The initial question is addressed: we show that the resulting boundary theory can itself be rewritten as an observer action, equivalently a $(0+1)$-dimensional HT theory. This yields a direct identification of the boundary clock with the $U(1)$ phase mode, and makes its relation to the bulk clock explicit.

Unimodular time in JT gravity: a holographic clock

Abstract

How is a ''bulk clock'' encoded holographically? We address this in Jackiw-Teitelboim (JT) gravity, where a natural physical clock emerges by promoting the vacuum energy to a dynamical variable: the vacuum cosmological constant becomes a top form degree of freedom conjugate to spacetime volume, thereby defining a notion of bulk physical time. This construction is naturally formulated in the Henneaux-Teitelboim (HT) framework. We show that the boundary dynamics is the Schwarzian mode coupled to a free particle on , matching the universal low-energy effective action of the complex SYK model. By further clarifying the role of the vacuum cosmological constant as a top form, we establish the equivalence between JT gravity coupled to two-dimensional Maxwell theory and 2d HT gravity via an explicit field redefinition. The initial question is addressed: we show that the resulting boundary theory can itself be rewritten as an observer action, equivalently a -dimensional HT theory. This yields a direct identification of the boundary clock with the phase mode, and makes its relation to the bulk clock explicit.
Paper Structure (35 sections, 166 equations, 3 figures)

This paper contains 35 sections, 166 equations, 3 figures.

Table of Contents

  1. Introduction and summary
  2. Background and motivation.
  3. Summary of results
  4. A two-dimensional HT gravity theory.
  5. Outline of the paper
  6. Euclidean HTHT2 gravity and complex SYK realisation
  7. JT gravity and the Schwarzian.
  8. Topological $U(1)$ sector.
  9. Variational principle for the HT term.
  10. Mixed boundary conditions and the complex SYK model
  11. Comment on counterterms.
  12. Thermal aspects of HTHT2 gravity
  13. Thermoelectric properties of the boundary theory
  14. HTHT2 gravity, 2d Maxwell and BF gauge theory
  15. Bridging 2d Maxwell to HTHT2
  16. ...and 20 more sections

Figures (3)

  • Figure 1: Panel (a) shows the coordinate systems on the hyperbolic disk. Panel (b) shows the standard deformation to the cigar geometry; the Schwarzian mode lives on the cutoff boundary, shown as the blue circle.
  • Figure 2: Left: the disk geometry, with constant-$\rho$ hypersurfaces $\Sigma_\rho$ highlighted in red. Right: the cigar geometry, in which unimodular time induces a preferred foliation, here taken to be the constant-$\rho$ slicing.
  • Figure 3: Here, we focus on the boundary, and the figure depicts how the bulk unimodular time relates to the boundary unimodular time.