We introduce the notion of higher Berry connection and curvature in the space of conformal boundary conditions in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi></mrow></math> conformal field theories, related to each other by exactly marginal boundary deformations, forming a “boundary conformal manifold.” Our definition builds upon previous works on tensor networks, such as matrix product states, where the triple inner product or multiwavefunction overlap plays a key geometric role. On the one hand, our boundary conformal field theory formulation of higher Berry phase provides a new analytic tool to study families of invertible phases in condensed matter systems. On the other hand, it uncovers a new geometric structure on the moduli space of conformal boundary conditions, beyond the usual Riemannian structure defined through the Zamolodchikov metric. When the boundary conformal manifold has an interpretation as the position moduli space of a D-brane, our higher Berry connection coincides with the Neveu-Schwarz-Neveu-Schwarz (NS-NS) B-field in string theory. The general definition does not require such an interpretation and is formulated purely field theoretically, in terms of correlation functions of boundary-condition-changing operators. We also explore a connection between higher Berry connections and functional Berry connections in the loop spaces of boundary conformal manifolds.

United States

Yichul Choi

Hyunsoo Ha

Dongyeob Kim

Japan

Yuya Kusuki

Austria

Shuhei Ohyama

Shinsei Ryu

Higher structures on boundary conformal manifolds: Higher Berry phase and boundary conformal field theory

We study the inverse problem of holographic entanglement entropy in AdS using a data-driven generative model. Training data consist of randomly generated geometries and their holographic entanglement entropies using the Ryu-Takayanagi formula. After training, the Transformer reconstructs the blackening function within our metric ansatz from previously unseen inputs. The Transformer achieves accurate reconstructions on smooth black hole geometries and extrapolates to horizonless backgrounds. We describe the architecture and data generation process, and we quantify accuracy on both $\textit{f}$($\textit{z}$) and the reconstructed $\textit{S}$($\textit{ℓ}$). Code and evaluation scripts are available at the provided repository.

South Korea

Sejin Kim

Learning the inverse Ryu-Takayanagi formula with transformers

We study the Ryu-Takayanagi (RT) surfaces associated with timelike subregions in static spacetimes with a horizon. It is possible to find the analytical continuation of the RT surfaces that can extend into the horizon, allowing us to probe the interior of the black hole. The horizon typically divides the RT surface into two distinct parts. We demonstrate that the area of the surface inside the horizon can be reconstructed from the contributions of the surfaces outside the horizon, along with additional RT surfaces for spacelike subregions that are causally related to the timelike subregions. This result provides a concrete realization of black hole complementarity at the level of classical metric, where the spacetime in the black hole interior can be reconstructed from the degrees of freedom outside the horizon.

China

Wu-zhong Guo

Jin Xu

Duality of Ryu-Takayanagi surfaces inside and outside the horizon

We study a class of quantum states involving multiple entangled CFTs in AdS /CFT , associated with multi-boundary black hole geometries, and demonstrate that the Ryu-Takayanagi (RT) formula for entanglement entropy can be derived using only boundary CFT data. Approximating the OPE coefficients by their Gaussian moments within the 2D large-$\textit{c}$ CFT ensemble, we show that both the norm of the states and the entanglement entropies associated with various bipartitions — reproducing the expected bulk dual results — can be computed purely from the CFT. All $\textit{macroscopic geometric}$ structures arising from gravitational saddles emerge entirely from the universal statistical moments of the $\textit{microscopic algebraic}$ CFT data, revealing a statistical-mechanical mechanism underlying semiclassical gravity. We establish a precise correspondence between the CFT norm, the Liouville partition function with ZZ boundary conditions, and the exact gravitational path integral over 3D multi-boundary black hole geometries. For entanglement entropy, each RT phase arises from a distinct leading-order Gaussian contraction, with phase transitions — analogous to replica wormholes — emerging naturally from varying dominant statistical patterns in the CFT ensemble. Our derivation elucidates how the general mechanism behind holographic entropy, namely a boundary replica direction that elongates and becomes contractible in the bulk dual, is encoded explicitly in the statistical structure of the CFT data.

Ning Bao

Hao Geng

Yikun Jiang

Ryu-Takayanagi formula for multi-boundary black holes from 2D large-$\textit{c}$ CFT ensemble

We study quantum quench processes in (1+1)-dimensional conformal field theory (CFT) in which the initial thermal equilibrium (Gibbs) state is time-evolved by spatially inhomogeneous Hamiltonians, the so-called Möbius and sine-square-deformed (SSD) Hamiltonians. We found that, when the quench is induced by the SSD Hamiltonian, almost all the degrees of freedom are asymptotically gathered at a single point, resulting in a point-like excitation. This excitation, which we dub black hole-like excitation, carries as much information as the total thermal entropy. In contrast, other parts of the system approach the low-entropy (low-temperature) state at late times. For the quench by the Möbius Hamiltonian, we instead found an eternal periodic oscillation of physical quantities such as von Neumann entropy for subsystems. When the CFT admits a holographic dual description, the SSD quench induces a time-dependent, inhomogeneous deformation of the bulk black hole horizon, which, at late enough times, “touches” the boundary. Our quench setups can be used as a way to create low-temperature states, and, also, simulate the formation and evaporation processes of black holes.

Kanato Goto

Masahiro Nozaki

Kotaro Tamaoka

Mao Tan

Non-equilibrating a black hole with inhomogeneous quantum quench

Using recent developments in expressing one-loop partition functions in Euclidean $\textit{AdS}$ space-times in terms of character integrals, we relate the one-loop effective action for a free field theory in $\textit{AdS}$ (comprised of a massless scalar field and a massless Majorana fermion field) to the partition function of the de Alfaro-Fubini-Furlan (DFF) conformal quantum mechanics (CQM) models on the two global $\textit{AdS}$ boundaries. The equal number of bosonic and fermionic degrees in the field theory guarantee that the one-loop calculation is free of all UV divergences except a logarithmic one consistent with the expected entanglement entropy behaviour in a CQM. Via a thermofield double representation, we compute the entanglement entropy between two copies of the $\textit{CFT}$ (CQM), each living near one of the two boundaries of global $\textit{AdS}$ , in a state at global time $\textit{τ}$ → – ∞. This entanglement entropy is expressed in terms of the logarithm of the regularised length of a closed particle trajectory infinitesimally near the rim of the Euclidean $\textit{AdS}$ disc. We view this relation between boundary quantum entanglement and a bulk geometrical quantity as the $\textit{AdS}$ /$\textit{CFT}$ version of the Ryu-Takayanagi conjecture in our setup. The boundary entanglement entropy is equal to 4 times the thermodynamic entropy read off from the regularised one-loop effective action in $\textit{AdS}$ . Further, we compute the bulk entanglement entropy associated with black hole horizons in Lorentzian $\textit{AdS}$ and show that it precisely matches the boundary entanglement entropy.

Portugal

Gabriel Cardoso

Bernardo Martins

Suresh Nampuri

Bulk-boundary entanglement correspondence and the Ryu-Takayanagi conjecture in an $\textit{AdS}$

Based on the renormalization group (RG) flow of worldsheet bosonic string theory, we construct an effective holographic dual description of the target space theory identifying the RG scale with the emergent extra dimension. This results in an effective dilaton-gravity-gauge theory, analogous to the low-energy description of bosonic M theory. We argue that this holographic dual effective field theory is nonperturbative in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>α</mi><mo>′</mo></msup></math> expansion, where a class of string quantum fluctuations are resummed to all orders. To investigate the monotonicity of the RG flow of the target space metric in the emergent spacetime, we consider entropy production along the RG flow. We construct a microscopic entropy functional based on the probability distribution function of the holographic dual effective field theory, regarded as Gibbs- or Shannon-type entropy. Given that the Ricci flow represents the 1-loop RG flow equation of the target space metric for the 2D nonlinear sigma model, and motivated by Perelman’s proof of the monotonicity of Ricci flow, we propose a Perelman’s entropy functional for the holographic dual effective field theory. This entropy functional is also nonperturbative in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>α</mi><mo>′</mo></msup></math> expansion, and thus, generalizes the 1-loop result to the all-loop order. Furthermore, utilizing the equivalence between the Hamilton-Jacobi equation and the local RG equation, we suggest that the RG flow of holographic Perelman’s entropy functional is the Weyl anomaly. This eventually reaffirms the monotonicity of RG flow for the emergent target spacetime but in a nonperturbative way. Interestingly, we find that the microscopic entropy production rate can be determined by integrating the rate of change of the holographic Perelman’s entropy functional over all possible metric configurations along the flow.

Ki-Seok Kim

Arpita Mitra

Debangshu Mukherjee

Monotonicity of the RG flow in an emergent dual holography of a worldsheet nonlinear <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>σ</mi></math> model

In this paper, our aim is to illustrate that the process of open string scattering corresponds to the evolution of the entanglement wedge, where the scattering distance is identified as the entanglement wedge cross section. Moreover, open-closed string scattering, specifically the disk-disk interaction, works for the evolution of the reflected entanglement wedge, with the circumference of the waist cross section equating to the reflected entropy. It therefore provides evidence for the deep connections between the string world sheet and the Ryu-Takayanagi surface. This connection is not only a coincidence rooted in hyperbolic geometry; it also reflects an additional correspondence between two distinct theories: mutual information and the geometric Batalin-Vilkoviski master equation.

Xin Jiang

Houwen Wu

Haitang Yang

String scattering and evolution of a Ryu-Takayanagi surface

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

Xi Dong

Donald Marolf

Zhencheng Wang

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

We establish an equivalence between two different quantum quench problems, the joining local quantum quench and the Möbius quench, in the context of (1 + 1)-dimensional conformal field theory (CFT). Here, in the former, two initially decoupled systems (CFTs) on finite intervals are joined at $\textit{t}$ = 0. In the latter, we consider the system that is initially prepared in the ground state of the regular homogeneous Hamiltonian on a finite interval and, after $\textit{t}$ = 0, let it time-evolve by the so-called Möbius Hamiltonian that is spatially inhomogeneous. The equivalence allows us to relate the time-dependent physical observables in one of these problems to those in the other. As an application of the equivalence, we construct a holographic dual of the Möbius quench from that of the local quantum quench. The holographic geometry involves an end-of-the-world brane whose profile exhibits non-trivial dynamics.

Jonah Kudler-Flam

Tokiro Numasawa

Bridging two quantum quench problems — local joining quantum quench and Möbius quench — and their holographic dual descriptions