On the construction of charged operators inside an eternal black hole

TL;DR

The paper shows that to reconstruct a charged bulk operator inside an eternal AdS black hole, one must introduce a boundary-to-boundary Wilson line $W_{LR}$ whose zero mode is physical only in two-sided geometries, making the operator state-dependent. It provides explicit $D=3$ Chern-Simons and Maxwell-based constructions, demonstrates locality of the Wilson line from the boundary CFT perspective, and connects the Wilson line to the thermofield double and to mirror operator frameworks. By analyzing Dirac brackets and OPEs, the work clarifies how the Wilson line encodes boundary charges and how it modifies standard HKLL-type reconstructions in a gauge-invariant manner. The findings have implications for understanding bulk locality, gauge-invariance in holography, and the state-dependence of interior operators, with potential extensions to gravitational Wilson lines and ER=EPR-inspired setups.

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.

Figures (11)

• Figure 2: The charged scalar connected to the right boundary via a Wilson line is a gauge-invariant bulk operator, carrying charges $Q_L=0$ and $Q_R =q$.
• Figure 3: Expression for the gauge-invariant bulk scalar $\hat{\phi}$ (blue line) in terms of the smeared dressed right operators (orange lines) and right-framed left operators (red lines). The left contributions can be decomposed into a dressed operator contribution and a boundary-to-boundary Wilson line.
• Figure 5: Wilson line stretching along a boundary-to-boundary geodesic.
• Figure 6: Path integral evaluation of the Wilson line action on the thermofield double state.
• Figure 7: Construction of the Wilson line via OPE fusion at the bifurcation surface of a negatively charged operator from the left and a positively charged operator from the right.