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Locality constraints in AdS$_2$ without parity

Manuel Loparco, Grégoire Mathys, João Penedones, Jiaxin Qiao, Xiang Zhao

TL;DR

This work identifies a fundamental subtlety in locality constraints for quantum field theories in AdS$_2$: the standard AdS$_{d+1}$ locality sum rules do not directly extend to $d=1$ due to the distinct boundary-ordering structure and the $ ho$-dependent conformal data. By combining unitarity with radial and equal-time quantization, the authors derive a uniform power-law bound and formulate dispersion relations that lead to a convergent decomposition of bulk-boundary-boundary correlators into even and odd local blocks. They establish two new families of locality sum rules, one symmetric and one antisymmetric, which hold without assuming parity, and they test these rules explicitly in the free scalar theory, verifying the even and odd sum rules in representative channels. The results provide a robust framework for constraining AdS$_2$ QFT data, enabling refined bootstrap-type analyses and applications to RG flows in AdS$_2$ and conformal line defects. The locality constraints thus offer a precise, parity-aware toolkit for probing locality and operator structure in one-dimensional CFT/AdS setups with potential broader impact on defect CFTs and holographic models.

Abstract

We study bulk locality constraints in quantum field theories in AdS$_2$. The known derivation of locality sum rules in AdS$_{d+1}$ does not apply for $d=1$ due to the different singularity structure of the conformal blocks and the inequivalence of operator orderings on the boundary. Assuming unitarity and a mild growth condition, we establish power-law bounds for correlators, derive dispersion relations and an expansion in terms of ``even'' and ``odd'' local blocks that converges in the entire AdS$_2$. These yield two novel families of symmetric and antisymmetric locality sum rules. We test these sum rules explicitly in the free scalar field theory.

Locality constraints in AdS$_2$ without parity

TL;DR

This work identifies a fundamental subtlety in locality constraints for quantum field theories in AdS: the standard AdS locality sum rules do not directly extend to due to the distinct boundary-ordering structure and the -dependent conformal data. By combining unitarity with radial and equal-time quantization, the authors derive a uniform power-law bound and formulate dispersion relations that lead to a convergent decomposition of bulk-boundary-boundary correlators into even and odd local blocks. They establish two new families of locality sum rules, one symmetric and one antisymmetric, which hold without assuming parity, and they test these rules explicitly in the free scalar theory, verifying the even and odd sum rules in representative channels. The results provide a robust framework for constraining AdS QFT data, enabling refined bootstrap-type analyses and applications to RG flows in AdS and conformal line defects. The locality constraints thus offer a precise, parity-aware toolkit for probing locality and operator structure in one-dimensional CFT/AdS setups with potential broader impact on defect CFTs and holographic models.

Abstract

We study bulk locality constraints in quantum field theories in AdS. The known derivation of locality sum rules in AdS does not apply for due to the different singularity structure of the conformal blocks and the inequivalence of operator orderings on the boundary. Assuming unitarity and a mild growth condition, we establish power-law bounds for correlators, derive dispersion relations and an expansion in terms of ``even'' and ``odd'' local blocks that converges in the entire AdS. These yield two novel families of symmetric and antisymmetric locality sum rules. We test these sum rules explicitly in the free scalar field theory.

Paper Structure

This paper contains 21 sections, 1 theorem, 109 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Bulk-boundary-boundary correlator in AdS2
  4. Geometry
  5. Embedding space
  6. Conformal block expansion
  7. Quantization and analyticity
  8. Analyticity from radial quantization
  9. Analyticity from equal-time quantization
  10. A uniform bound
  11. Locality constraints
  12. Dispersion relation for the full correlator
  13. Local block expansion
  14. Locality sum rules
  15. Comments on conformal line defects
  16. ...and 6 more sections

Key Result

Theorem 2.1

(Phragmén–Lindelöf theorem, sector version) Let $f(z)$ be an analytic function in the sector domain and continuous up to its boundary. Suppose that $f(z)$ is uniformly bounded along the boundary of the sector: and that, as $r\rightarrow\infty$, uniformly in $\arg(z)$ for some finite $A$ and some $\beta<\tfrac{\pi}{\theta_2-\theta_1}$. Then for all $z\in\mathcal{S}_{\theta_1,\theta_2}$.

Figures (7)

  • Figure 1: The range of $\rho$ depending on the position of the bulk point. Here the two boundary points are fixed. Exchanging the positions of $\mathcal{O}_i$ and $\mathcal{O}_j$ leads to $\rho\rightarrow-\rho$.
  • Figure 2: Conformal block expansion for different types of configurations. The choice of $C_{ijl}$ vs $C_{jil}$ depends on the cyclic ordering of the boundary operators. For figure \ref{['fig:bkbdbd1']}, the cyclic ordering is $[lij]=[ijl]=[jli]$, so the corresponding OPE coefficient is $C_{ijl}$. While for figure \ref{['fig:bkbdbd2']}, the corresponding OPE coefficient is $C_{jil}$.
  • Figure 3: Three types of quantizations in the upper half plane (upper row) and in the Poincaré disk (lower row): $(a)$ Radial quantization; $(b)$ North-South quantization; $(c)$ Equal-time quantization.
  • Figure 4: Contour deformation.
  • Figure 5: Integration contours $C$ (blue) and $C'$ (orange) of the integral transform \ref{['localblock:int_rep']}. The poles starting from $\Delta_{ij}$ and $\Delta_{ji}$ belong to the even local block integral representation while those starting from $\Delta_{ij}-1$ and $\Delta_{ji}-1$ belong to the odd one. These poles are shifted along the imaginary axis to avoid cluttering. Notice the origin is at $\Delta'=\tfrac{1}{2}$.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Theorem 2.1