A correction to this paper has been published: https://doi.org/10.1007/JHEP04(2025)171

United States

Molly Kaplan

Donald Marolf

Xuyang Yu

Ying Zhao

Erratum to: De Sitter quantum gravity and the emergence of local algebras

Previous studies of linearized stability of asymptotically flat Euclidean axion wormholes found that symmetric modes suffered from divergences. We show that such divergences were an artifact of a particular way of solving the constraints, and that a full treatment leads to finite actions for such modes. The modes must thus be included in a stability analysis. However, since the action for these modes turns out to be positive, this turns out not to affect previous statements about stability of axion wormholes. We also introduce a technique that allows us to show this positivity at a pseudo-analytic level that avoids heavy numerics. Our techniques should be useful to future studies of stabilities of other wormholes as well.

Italy

Bruno Missoni

Euclidean wormholes stability analysis revisited

The Euclidean Einstein-Hilbert action is well-known to be unbounded below and thus to raise many questions regarding the definition of the gravitational path integral. A variety of works since the late 1980’s have suggested that this problem disappears when one fixes a foliation of the spacetime and imposes the corresponding gravitational constraints. However, we show here that this approach fails with various classes of boundary conditions imposed on the foliation: compact slices without boundary, asymptotically flat, or asymptotically locally anti-de Sitter slices. We also discuss the idea of fixing the scalar curvature and Wick-rotating the conformal factor, and show that it also fails to produce an action bounded from below.

Gary Horowitz

United Kingdom

Jorge Santos

Constraints are not enough

In holographic theories, the Hubeny-Rangamani-Takayanagi (HRT) area operator plays a key role in our understanding of the emergence of semiclassical Einstein-Hilbert gravity. When higher derivative corrections are included, the role of the area is instead played by a more general functional known as the geometric entropy. It is thus of interest to understand the flow generated by the geometric entropy on the classical phase space. In particular, the fact that the associated flow in Einstein-Hilbert or Jackiw-Teitelboim (JT) gravity induces a relative boost between the left and right entanglement wedges is deeply related to the fact that gravitational dressing promotes the von Neumann algebra of local fields in each wedge to type II. This relative boost is known as a boundary-condition-preserving (BCP) kink-transformation. In a general theory of gravity (with arbitrary higher-derivative terms), it is straightforward to show that the flow continues to take the above geometric form when acting on a spacetime where the HRT surface is the bifurcation surface of a Killing horizon. However, the form of the flow on other spacetimes is less clear.

Xi Dong

Pratik Rath

Geometric entropies and their Hamiltonian flows

Quantum theories of gravity are generally expected to have some degree of non-locality, with familiar local physics emerging only in a particular limit. Perturbative quantum gravity around backgrounds with isometries and compact Cauchy slices provides an interesting laboratory in which this emergence can be explored. In this context, the remaining isometries are gauge symmetries and, as a result, gauge-invariant observables cannot be localized. Instead, local physics can arise only through certain relational constructions.

De Sitter quantum gravity and the emergence of local algebras

Near-extremal black holes are subject to large quantum effects, which modify their low-temperature thermodynamic behavior. Hitherto, these quantum effects were analyzed by separating the geometry into the near-horizon region and its exterior. It is desirable to understand and reproduce such corrections from the full higher-dimensional asymptotically flat or AdS geometry’s perspective. We address this question in this article and fill this gap. Specifically, we find off-shell eigenmodes of the quadratic fluctuation operator of the Euclidean gravitational dynamics, with eigenvalues that vanish linearly with temperature. We illustrate this for BTZ and neutral black holes with hyperbolic horizons in AdS in Einstein-Hilbert theory, and for the charged black holes in Einstein-Maxwell theory. The linear scaling with Matsubara frequency, which is a distinctive feature of the modes, together with the fact that their wavefunctions localize close to the horizon as we approach extremality, identifies them as responsible for the aforementioned quantum effects. We provide a contour prescription to deal with the sign indefiniteness of the Euclidean Einstein-Maxwell action, which we derive to aid our analysis. We also resolve a technical puzzle regarding modes associated with rotational isometries in stationary black hole spacetimes.

Maciej Kolanowski

Ilija Rakic

Mukund Rangamani

Gustavo Turiaci

Looking at extremal black holes from very far away

Gravitational Rényi computations have traditionally been described in the language of Euclidean path integrals. In the semiclassical limit, such calculations are governed by Euclidean (or, more generally, complex) saddle-point geometries. We emphasize here that, at least in simple contexts, the Euclidean approach suggests an alternative formulation in terms of the bulk quantum wavefunction. Since this alternate formulation can be directly applied to the real-time quantum theory, it is insensitive to subtleties involved in defining the Euclidean path integral. In particular, it can be consistent with many different choices of integration contour.

Jesse Held

Xiaoyi Liu

Zhencheng Wang

Euclidean and complex geometries from real-time computations of gravitational Rényi entropies

Recent works by Chandrasekaran, Penington, and Witten have shown in various special contexts that the quantum-corrected Ryu-Takayanagi (RT) entropy (or its covariant Hubeny-Rangamani-Takayanagi (HRT) generalization) can be understood as computing an entropy on an algebra of bulk observables. These arguments do not rely on the existence of a local holographic dual field theory. We show that analogous-but-stronger results hold in any UV-completion of asymptotically anti-de Sitter quantum gravity with a Euclidean path integral satisfying a simple and (largely) familiar set of axioms. We consider a quantum context in which a standard Lorentz-signature classical bulk limit would have Cauchy slices with asymptotic boundaries B  L  ⊔ B  R  where both B  L  and B  R  are compact manifolds without boundary. Our main result is then that (the UV-completion of) the quantum gravity path integral defines type I von Neumann algebras   <math> <msubsup> <mi>A</mi> <mi>L</mi> <msub> <mi>B</mi> <mi>L</mi> </msub> </msubsup> </math>  $$ {\mathcal{A}}_L^{B_L} $$ ,   <math> <msubsup> <mi>A</mi> <mi>R</mi> <msub> <mi>B</mi> <mi>R</mi> </msub> </msubsup> </math>  $$ {\mathcal{A}}_R^{B_R} $$  of observables acting respectively at B  L , B  R  such that   <math> <msubsup> <mi>A</mi> <mi>L</mi> <msub> <mi>B</mi> <mi>L</mi> </msub> </msubsup> </math>  $$ {\mathcal{A}}_L^{B_L} $$ ,   <math> <msubsup> <mi>A</mi> <mi>R</mi> <msub> <mi>B</mi> <mi>R</mi> </msub> </msubsup> </math>  $$ {\mathcal{A}}_R^{B_R} $$  are commutants. The path integral also defines entropies on   <math> <msubsup> <mi>A</mi> <mi>L</mi> <msub> <mi>B</mi> <mi>L</mi> </msub> </msubsup> </math>  $$ {\mathcal{A}}_L^{B_L} $$ ,   <math> <msubsup> <mi>A</mi> <mi>R</mi> <msub> <mi>B</mi> <mi>R</mi> </msub> </msubsup> </math>  $$ {\mathcal{A}}_R^{B_R} $$ . Positivity of the Hilbert space inner product then turns out to require the entropy of any projection operator to be quantized in the form ln N for some N ∈ ℤ + (unless it is infinite). As a result, our entropies can be written in terms of standard density matrices and standard Hilbert space traces. Furthermore, in appropriate semiclassical limits our entropies are computed by the RT-formula with quantum corrections. Our work thus provides a Hilbert space interpretation of the Ryu-Takayanagi entropy. Since our axioms do not severely constrain UV bulk structures, it is plausible that they hold equally well for successful formulations of string field theory, spin-foam models, or any other approach to constructing a UV-complete theory of gravity.

Eugenia Colafranceschi

Algebras and Hilbert spaces from gravitational path integrals. Understanding Ryu-Takayanagi/HRT as entropy without AdS/CFT

Spacetime wormholes can provide non-perturbative contributions to the gravitational path integral that make the actual number of states e  S  in a gravitational system much smaller than the number of states   <math> <msup> <mi>e</mi> <msub> <mi>S</mi> <mi>p</mi> </msub> </msup> </math>  $$ {e}^{S_{\textrm{p}}} $$  predicted by perturbative semiclassical effective field theory. The effects on the physics of the system are naturally profound in contexts in which the perturbative description actively involves N = O(e  S ) of the possible   <math> <msup> <mi>e</mi> <msub> <mi>S</mi> <mi>p</mi> </msub> </msup> </math>  $$ {e}^{S_{\textrm{p}}} $$  perturbative states; e.g., in late stages of black hole evaporation. Such contexts are typically associated with the existence of non-trivial quantum extremal surfaces. However, by forcing a simple topological gravity model to evolve in time, we find that such effects can also have large impact for N ≪ e  S  (in which case no quantum extremal surfaces can arise). In particular, even for small N, the insertion of generic operators into the path integral can cause the non-perturbative time evolution to differ dramatically from perturbative expectations. On the other hand, this discrepancy is small for the special case where the inserted operators are non-trivial only in a subspace of dimension D ≪ e  S . We thus study this latter case in detail. We also discuss potential implications for more realistic gravitational systems.

Poland

Null states and time evolution in a toy model of black hole dynamics

Euclidean path integrals for UV-completions of d-dimensional bulk quantum gravity were recently studied in [1] by assuming that they satisfy axioms of finiteness, reality, continuity, reflection-positivity, and factorization. Sectors   <math> <msub> <mi>H</mi> <mi>B</mi> </msub> </math>  $$ {\mathcal{H}}_{\mathcal{B}} $$  of the resulting Hilbert space were then defined for any (d − 2)-dimensional surface   <math> <mi>B</mi> </math>  $$ \mathcal{B} $$ , where   <math> <mi>B</mi> </math>  $$ \mathcal{B} $$  may be thought of as the boundary ∂Σ of a bulk Cauchy surface in a corresponding Lorentzian description, and where   <math> <mi>B</mi> </math>  $$ \mathcal{B} $$  includes the specification of appropriate boundary conditions for bulk fields. Cases where   <math> <mi>B</mi> </math>  $$ \mathcal{B} $$  was the disjoint union B ⊔ B of two identical (d − 2)-dimensional surfaces B were studied in detail and, after the inclusion of finite-dimensional ‘hidden sectors,’ were shown to provide a Hilbert space interpretation of the associated Ryu-Takayanagi entropy. The analysis was performed by constructing type-I von Neumann algebras   <math> <msubsup> <mi>A</mi> <mi>L</mi> <mi>B</mi> </msubsup> </math>  $$ {\mathcal{A}}_L^B $$ ,   <math> <msubsup> <mi>A</mi> <mi>R</mi> <mi>B</mi> </msubsup> </math>  $$ {\mathcal{A}}_R^B $$  that act respectively at the left and right copy of B in B ⊔ B.