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A One-Loop Test of the near-AdS$_2$/near-CFT$_1$ Correspondence

Anthony M. Charles, Finn Larsen

TL;DR

This work validates near-AdS$_2$/near-CFT$_1$ holography by performing a bulk one-loop quantization of JT gravity and matching the Schwarzian/SYK logarithmic correction. The key insight is that, after separating continuous and discrete modes on AdS$_2$, the logarithmic term in the partition function is entirely due to discrete quadratic holomorphic differentials, yielding $\log Z|_{\text{one-loop}} = \frac{3}{2}\log\frac{G_2 T}{M_{\text{gap}}} = -\frac{3}{2}\log\frac{\beta M_{\text{gap}}}{G_2}$, precisely reproducing the SYK result. The analysis also shows that continuous modes and harmonic vectors do not generate logarithms, and that one-loop corrections to entropy vanish, so the microcanonical entropy remains $S_{\text{BH}} = \frac{\Phi|_H}{4G_2}$. The results imply a universal structure for near-extremal black holes and offer a bulk-rooted alternative to Schwarzian-boundary approaches for understanding near-AdS$_2$ holography.

Abstract

We analyze quantum fluctuations around black hole solutions to the Jackiw-Teitelboim model. We use harmonic analysis on Euclidean AdS$_2$ to show that the logarithmic corrections to the partition function are determined entirely by quadratic holomorphic differentials, even when conformal symmetry is broken and harmonic modes are no longer true zero modes. Our quantum-corrected partition function agrees precisely with the SYK result. We argue that our effective quantum field theory methods and results generalize to other theories of two-dimensional dilaton gravity.

A One-Loop Test of the near-AdS$_2$/near-CFT$_1$ Correspondence

TL;DR

This work validates near-AdS/near-CFT holography by performing a bulk one-loop quantization of JT gravity and matching the Schwarzian/SYK logarithmic correction. The key insight is that, after separating continuous and discrete modes on AdS, the logarithmic term in the partition function is entirely due to discrete quadratic holomorphic differentials, yielding , precisely reproducing the SYK result. The analysis also shows that continuous modes and harmonic vectors do not generate logarithms, and that one-loop corrections to entropy vanish, so the microcanonical entropy remains . The results imply a universal structure for near-extremal black holes and offer a bulk-rooted alternative to Schwarzian-boundary approaches for understanding near-AdS holography.

Abstract

We analyze quantum fluctuations around black hole solutions to the Jackiw-Teitelboim model. We use harmonic analysis on Euclidean AdS to show that the logarithmic corrections to the partition function are determined entirely by quadratic holomorphic differentials, even when conformal symmetry is broken and harmonic modes are no longer true zero modes. Our quantum-corrected partition function agrees precisely with the SYK result. We argue that our effective quantum field theory methods and results generalize to other theories of two-dimensional dilaton gravity.

Paper Structure

This paper contains 26 sections, 147 equations, 2 figures.

Table of Contents

  1. Introduction and Summary
  2. The Jackiw-Teitelboim Model
  3. Action and Equations of Motion
  4. Black Hole Solutions
  5. The Semi-Classical, Near-Extremal Regime
  6. Global Coordinates, Killing Vectors, and Conformal Symmetry
  7. Quantizing Fields in AdS2
  8. Metrics and Normalizations
  9. Scalar Fields in AdS2
  10. Dualization of Vectors and Tensors to Scalars
  11. AdS2 as a Worldsheet
  12. Free Field Partition Function
  13. Quantum Corrections to the Jackiw-Teitelboim Model
  14. Effective Field Theory Expectations
  15. One-Loop Quadratic Action
  16. ...and 11 more sections

Figures (2)

  • Figure 1: A schematic diagram of the hierarchy of different scales in our setup. This choice in scales ensures that the Jackiw-Teitelboim model is an effective field theory that can describe semi-classical, near-extremal black holes.
  • Figure 2: The spectrum of various types of Euclidean AdS$_2$ field modes with an on-shell mass of $m^2 = \frac{2}{\ell^2}$. The quadratic holomorphic differentials are zero modes in the extremal limit, so they dominate finite-temperature effects.