Diving into booklet wormholes

Abstract

ArXiv:2508.17898 proposed the booklet wormhole as the holographic dual of the GHZ state. This paper extends the investigation into this geometry, particularly focusing on the junction conditions for matter fields. We show that the symmetry of the GHZ state requires the bulk to admit special Killing vector fields that standard manifolds cannot realize. Moreover, these bulk symmetries require unprecedented quantum non-local junction conditions at the multi-way interface: Observers entering from different horizons will perceive different states inside the wormhole, where the junction conditions appear as constraints on the observables of different sets of observers. We finally discuss how to render booklet wormholes traversable via boundary deformations. A localized wave packet injected from one page generally evolves into a non-local mixed state on each remaining page, with the information encoded in the entanglement between different pages.

Figures (6)

• Figure 1: Both panels illustrate the Euclidean preparation of a Lorentzian geometry, where the bottom and top halves correspond to the Euclidean and Lorentzian sections, respectively. The left panel shows the Euclidean preparation of the canonical TFD state \ref{['theTFD']} dual to a standard two-sided black hole, where $\ket{\text{EPR}}\propto\sum_i\ket i (\Theta \ket i)$. The right panel illustrates the preparation of a generic TFD state, where $\ket {{\text{EPR}'}}\propto\sum_i\ket i (U\Theta \ket i)$. The dual geometry remains a regular two-sided black hole but features a topological interface inserted at the center (depicted by the red dash--dotted line), which is transparent to traversing quanta.
• Figure 2: Illustration of the topological property of the GHZ junction. Here, $a,b,c$ and $a',b',c'$ denote the lengths of the imaginary-time segments for each leg. The path integral yields an identical quantum state provided that $a+b+c = a'+b'+c'$.
• Figure 3: Illustration of a method for computing a tripartite correlator in a thermal GHZ state. The left panel depicts the path integral for a general tripartite correlation function, where three operators are inserted on their respective pages at the $t=0$ symmetric plane. Exploiting the topological property of the junction, the path integral is deformed into the configuration shown in the right panel. By representing the combined effect of $O_2$ and $O_3$ as an effective operator $O_{\text{eff}}$, the GHZ three-point function is mapped onto a two-point function of $O_1$ and $O_{\text{eff}}$ in a TFD state.
• Figure 4: Illustration of symmetries inside the horizons. Three pages are connected by a multi-way junction at the center. For any observer, the spacetime is smooth at the topological junction; therefore, we use dotted lines to indicate that each page can be extended to infinity. The geometry is symmetric under three multi-boosts. Each multi-boost translates the spacelike coordinate $t$ on two pages while leaving the remaining page invariant.
• Figure 5: First thought experiment. The upper and lower Penrose diagrams each represent a booklet wormhole, glued along a multi-way junction indicated by the dash-dotted line. In the upper panel, three particles fall into the black holes from the three boundaries, appearing to meet at a spacetime point on the junction. The lower panel shows the same situation after a multi-boost. After the transformation, the particles on Pages 1 and 2 hit the singularity before reaching the junction, suggesting that these two particles can never meet.