On the construction of charged operators inside an eternal black hole

TL;DR

This paper explores the holographic construction of charged bulk operators inside an eternal AdS black hole, revealing the necessity of Wilson lines connecting boundaries and their state-dependent nature, with explicit results in three dimensions.

Contribution

It introduces a Wilson line-based method for constructing charged operators in eternal black holes, highlighting their state dependence and providing explicit three-dimensional examples.

Findings

Wilson lines are essential for charged operator construction in eternal black holes.

The Wilson line acts as a local operator in either boundary CFT.

Explicit expressions for charged operators and Wilson lines in three dimensions.

Abstract

We revisit the holographic construction of (approximately) local bulk operators inside an eternal AdS black hole in terms of operators in the boundary CFTs. If the bulk operator carries charge, the construction must involve a qualitatively new object: a Wilson line that stretches between the two boundaries of the eternal black hole. This operator - more precisely, its zero mode - cannot be expressed in terms of the boundary currents and only exists in entangled states dual to two-sided geometries, which suggests that it is a state-dependent operator. We determine the action of the Wilson line on the relevant subspaces of the total Hilbert space, and show that it behaves as a local operator from the point of view of either CFT. For the case of three bulk dimensions, we give explicit expressions for the charged bulk field and the Wilson line. Furthermore, we show that when acting on the thermofield double state, the Wilson line may be extracted from a limit of certain standard CFT operator expressions. We also comment on the relationship between the Wilson line and previously discussed mirror operators in the eternal black hole.