An Exact Operator That Knows Its Location

TL;DR

This work constructs an exact bulk proto-field $\phi$ in $AdS_3$/CFT$_2$ by expressing it as an infinite Virasoro descendant expansion of a CFT$_2$ primary $\mathcal{O}$, with two complementary routes: a gravitational Wilson line (bulk-boundary OPE block) approach and a symmetry-based algebraic definition. The authors show that $\phi$ transforms as a bulk scalar under Virasoro diffeomorphisms and that its vacuum correlators reproduce bulk perturbation theory at large $c$, while remaining valid in general vacuum AdS$_3$ geometries, including BTZ black holes, enabling potentially non-perturbative probes of bulk locality and horizons. They provide an exact algebraic construction for $\phi$, derive explicit low-order solutions and a general framework in terms of orthogonal quasi-primaries, and establish a recursion relation for stress-tensor correlators that reproduces $\langle \phi\mathcal{O}T\rangle$, $\langle \phi\mathcal{O}TT\rangle$, and $\langle \phi\mathcal{O}T\bar T\rangle$. These results offer a non-perturbative handle on bulk reconstruction in $AdS_3$/CFT$_2$, with potential applications to bulk locality and horizon physics, and lay groundwork for extensions to more general holographic settings.

Abstract

We use conformal symmetry to define an AdS$_3$ proto-field $φ$ as an exact linear combination of Virasoro descendants of a CFT$_2$ primary operator $\mathcal{O}$. We find that both symmetry considerations and a gravitational Wilson line formalism lead to the same results. The operator $φ$ has many desirable properties; in particular it has correlators that agree with gravitational perturbation theory when expanded at large $c$, and that automatically take the correct form in all vacuum AdS$_3$ geometries, including BTZ black hole backgrounds. In the future it should be possible to use $φ$ to probe bulk locality and black hole horizons at a non-perturbative level.

Figures (2)

• Figure 1: This figure portrays a bulk-boundary OPE block used to compute the correlator (\ref{['eq:phiOTTb']}). The red line denotes the gravitational or Chern-Simons Wilson line, while the blue circle suggests radial quantization around the block, so that it creates a definite linear combination of Virasoro descendants of the identity. The explicit computation involves line integrals over stress tensor correlators.
• Figure 2: Dashed (solid) lines are graviton (scalar) propagators.