Global symmetry, Euclidean gravity, and the black hole information problem

TL;DR

This work argues that unitary black hole evaporation compatible with the Bekenstein–Hawking entropy formula $S_{BH}=\frac{\mathrm{Area}}{4G}$ forbids global symmetries in quantum gravity, extending the Page-curve/QES framework beyond AdS/CFT. By analyzing low-dimensional models that admit global symmetries and contrasting them with holographic setups, the authors propose that Euclidean quantum gravity is deeply linked to holography. Central to the argument are quantum extremal surfaces, islands, and replica wormholes, which together yield unitary Page curves and constrain UV completions. The paper further contends that the Euclidean path integral appropriately computes entropies only in holographic theories, suggesting a fundamental equivalence between Euclidean gravity and holography in realistic quantum gravity.

Abstract

In this paper we argue for a close connection between the non-existence of global symmetries in quantum gravity and a unitary resolution of the black hole information problem. In particular we show how the essential ingredients of recent calculations of the Page curve of an evaporating black hole can be used to generalize a recent argument against global symmetries beyond the AdS/CFT correspondence to more realistic theories of quantum gravity. We also give several low-dimensional examples of quantum gravity theories which do not have a unitary resolution of the black hole information problem in the usual sense, and which therefore can and do have global symmetries. Motivated by this discussion, we conjecture that in a certain sense Euclidean quantum gravity is equivalent to holography.

Figures (5)

• Figure 1: The evaporating wormhole in JT gravity studied in Almheiri:2019psf. Turning on the interaction between the green gravitational region and the grey reservoir region produces two positive energy shells, shown as orange dashed lines. After this, positive energy Hawking radiation (shown in blue) leaks out into the reservoir system, while negative energy radiation (shown in red) falls into the right black hole and gradually decreases its energy.
• Figure 2: The new quantum extremal surface $\gamma$ of Almheiri:2019psfPenington:2019npb for an evaporating JT wormhole. The entanglement wedge of the reservoir at the boundary time indicated by the red dot is shaded yellow, note in particular the island in the gravitational region.
• Figure 3: Two situations where a bulk global symmetry would lead to a contradiction in AdS/CFT (figures from Harlow:2018tng). The green circle represents a putative bulk operator which is charged under the global symmetry, and the dotted line represents its (neutral) gravitational dressing. The contradiction is that in the boundary CFT a global symmetry operator $U(g)$ must be a product of operators whose support is each at most one of the $R_i$, and each of those operators can therefore only act nontrivially on bulk operators in the entanglement wedge of that $R_i$. Thus none of the operators, and therefore also their product, can act nontrivially on the green charged operator: it must be neutral.
• Figure 4: The semiclassical picture of a situation where a global symmetry in a quantum gravity theory with unitary black hole evaporation leads to a contradiction. The solid red line indicates a Cauchy slice of the reservoir system $R$, the red dot indicates the quantum gravity system $S$, the black dot is the quantum extremal surface $\gamma$ found in Almheiri:2019psfPenington:2019npb, the vertical dashed line is the boundary between the gravitational and non-gravitational parts of the semiclassial description, and $R_1, R_2,\ldots$ are subregions of $R$. The entanglement wedges of the $R_i$ are shaded grey, the entanglement wedge of $S$ is shaded pink, and the "island" which is part of the entanglement wedge of $R$ is shaded yellow. The green dot represents an operator which is charged under the putative global symmetry. Since the global symmetry operator $U(g)$ is a product of an operator on $S$ and operators on the $R_i$, none of its pieces can act nontrivially on the operator at the green dot and thus that operator must be neutral.
• Figure 5: Two contributions to the Euclidean gravity path integral with a boundary thermal circle of circumference $\beta$. On the left some cycle of the transverse directions contracts at the dotted line, while on the right it is the thermal circle which contracts. The contribution on the left is what is obtained from canonical quantization of gravitational effective field theory, and gives no contribution to the entropy at leading order in the gravitational coupling. The contribution on the right leads to the black hole entropy \ref{['BHform']}; it should be included only in effective theories which are UV-completed into a holographic description.