A CFT Perspective on Gravitational Dressing and Bulk Locality

TL;DR

<3-5 sentence high-level summary> Reconstructing local bulk operators in AdS/CFT with gravitational dressing, the paper develops a CFT-based framework in AdS_3/CFT_2 where bulk operators are realized as Virasoro cross-cap states. Gravitational dressing is encoded by a dressing operator and encoded geometrically as a sum over bulk geometries, yielding background independence and micro-causality at leading order in large N. The authors show that Virasoro (and, when present, Kac-Moody) symmetry eliminates non-local branch cuts in correlators and that a holographic bootstrap constraint suffices to reproduce HKLL-type locality restoration for interacting fields. The framework connects uniformization, soft-graviton Ward identities, and the bulk equation of motion, providing a concrete CFT mechanism for gravitational dressing and a path toward bulk locality beyond the strict bulk semiclassical limit.

Abstract

We revisit the construction of local bulk operators in AdS/CFT with special focus on gravitational dressing and its consequences for bulk locality. Specializing to 2+1-dimensions, we investigate these issues via the proposed identification between bulk operators and cross-cap boundary states. We obtain explicit expressions for correlation functions of bulk fields with boundary stress tensor insertions, and find that they are free of non-local branch cuts but do have non-local poles. We recover the HKLL recipe for restoring bulk locality for interacting fields as the outcome of a natural CFT crossing condition. We show that, in a suitable gauge, the cross-cap states solve the bulk wave equation for general background geometries, and satisfy a conformal Ward identity analogous to a soft graviton theorem, Virasoro symmetry, the large N conformal bootstrap and the uniformization theorem all play a key role in our derivations.

Figures (7)

• Figure 1: A bulk point $X$ in AdS lies at the intersection of a continuous family of geodesics, and thereby induces an antipodal pairing between boundary points.
• Figure 2: The limits $\eta \to 0$ and $\eta\to1$ of the bulk-to-boundary three-point function $\bigl\langle\Phi{\cal O}_i{\cal O}_j\bigr \rangle$ correspond to the two OPE limits where ${\cal O}_i$ approaches ${\cal O}_j$ or its mirror image nonocft. For physical cross caps both limits correspond to the same physical situation. For holographic cross caps, bulk locality requires that the limit $\eta\to1$ is regular.
• Figure 3: Correlation functions with cross-cap operators are most easily computed by the method of images, by associating to every local operator ${\cal O}(z)$ a ${\mathbb Z}_2$ image operator ${\cal O}'(z')$ placed at the antipodal point $z' = x - \frac{y^2}{\bar{z}-\bar{x}}$.
• Figure 4: Deformation of the circle along which the cross-cap identification is made.
• Figure 5: The uniformization theorem provides a map from a bulk-to-boundary 2-point function $\bigl\langle \Omega\bigl|\space \Phi(g) {\cal O}(x_1)\bigl|\space \Omega \bigr \rangle$ in a Bañados geometry to a 2-point function $\bigl\langle 0 \bigl|\space \Phi(X) {\cal O}(x_1)\bigl|\space 0 \bigr \rangle$ in AdS${}_3$. The bulk point $X_0$ depends on both $g_0$ and $\Omega$. The map makes essential use of Virasoro symmetry.