An exact operator that knows its location

Abstract

We use conformal symmetry to define an AdS<sub>3</sub> proto-field φ as an exact linear combination of Virasoro descendants of a CFT<sub>2</sub> primary operator <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </math>. 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<sub>3</sub> 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.