# Abdalla, Antonini, Iliesiu, Levine ## The gravitational path integral from an observer's point of view \[Links: [arXiv](https://arxiv.org/abs/2501.02632), [PDF](https://arxiv.org/pdf/2501.02632)\] \[Abstract: One of the fundamental problems in quantum gravity is to describe the experience of a gravitating observer in generic spacetimes. In this paper, we develop a framework for describing non-perturbative physics relative to an observer using the [[0555 Gravitational path integral|gravitational path integral]]. We apply our proposal to an observer that lives in a [[0632 Closed universe|closed universe]] and one that falls behind a black hole horizon. We find that the Hilbert space that describes the experience of the observer is much larger than the Hilbert space in the absence of an observer. In the case of closed universes, the Hilbert space is not one-dimensional, as calculations in the absence of the observer suggest. Rather, its dimension scales exponentially with $G_N^{-1}$. Similarly, from an observer's perspective, the dimension of the Hilbert space in a two-sided black hole is increased. We compute various observables probing the experience of a gravitating observer in this Hilbert space. We find that an observer experiences non-trivial physics in the closed universe in contrast to what it would see in a one-dimensional Hilbert space. In the two-sided black hole setting, our proposal implies that non-perturbative corrections to effective field theory for an infalling observer are suppressed until times exponential in the black hole entropy, resolving a recently raised puzzle in black hole physics. While the framework that we develop is exemplified in the toy-model of [[0050 JT gravity|JT gravity]], most of our analysis can be extended to higher dimensions and, in particular, to generic spacetimes not admitting a conventional holographic description, such as cosmological universes or black hole interiors.\] # Agrawal, Charalambous, Donnay ## Null infinity as an inverted extremal horizon: Matching an infinite set of conserved quantities for gravitational perturbations \[Links: [arXiv](https://arxiv.org/abs/2506.15526), [PDF](https://arxiv.org/pdf/2506.15526)\] \[Abstract: Every spacetime that is asymptotically flat near null infinity can be conformally mapped via a spatial inversion onto the geometry around an extremal, non-rotating and non-expanding horizon. We set up a dictionary for this geometric duality, connecting the geometry and physics near null infinity to those near the dual horizon. We then study its physical implications for conserved quantities for extremal black holes, extending previously known results to the case of gravitational perturbations. In particular, we derive a tower of near-horizon gravitational charges that are exactly conserved and show their one-to-one matching with [[0456 Newman-Penrose charges|Newman-Penrose conserved quantities]] associated with gravitational perturbations of the extremal Reissner-Nordström black hole geometry. We furthermore demonstrate the physical relevance of spatial inversions for extremal Kerr-Newman black holes, even if the latter are notoriously not conformally isometric under such inversions.\] # Akers, Bueller, DeWolfe, Higginbotham, Reinking, Rodriguez ## On observers in holographic maps \[Links: [arXiv](https://arxiv.org/abs/2503.09681), [PDF](https://arxiv.org/pdf/2503.09681)\] \[Abstract: A straightforward [[0555 Gravitational path integral|gravitational path integral]] calculation implies that [[0632 Closed universe|closed universes]] are trivial, described by a one dimensional Hilbert space. Two recent papers by [[2025#Harlow, Usatyuk, Zhao|Harlow-Usatyuk-Zhao]] and [[2025#Abdalla, Antonini, Iliesiu, Levine|Abdalla-Antonini-Iliesiu-Levine]] have sought to ameliorate this issue by defining special rules to incorporate observers into the path integral. However, the proposed rules are different, leading to differing results for the Hilbert space dimension. Moreover, the former work offers a holographic map realized using a [[0576 Non-isometric codes|non-isometric code]] construction to complement their path integral result and clarify its physics. In this work, we propose a non-isometric code that implements the second construction, allowing thorough comparison. Our prescription may be thought of as simply removing the portion of the map that acts on the observer, while preserving the rest, creating an effective holographic boundary at the observer-environment interface. This proposal can be directly applied to general holographic maps for both open and closed universes of any dimension.\] # Akhoury, Schutz, Garfinkle ## Superrotations are Linkages \[Links: [arXiv](https://arxiv.org/abs/2507.04245), [PDF](https://arxiv.org/pdf/2507.04245)\] \[Abstract: We show that superrotations can be described using the geometric conformal completion method of Penrose. In particular, superrotation charges can be described and calculated using the linkage method of Geroch and Winicour. Whether superrotation charges are calculated using the coordinate based Bondi formalism or the geometric Penrose formalism, the fact that the superrotation blows up at a point makes the superrotation charge formally ill defined. Nonetheless, we show that it can be made well defined through a regularization procedure devised by Flanagan and Nichols.\] # Almheiri ## Measurements *with* probabilities in the final state proposal \[Links: [arXiv](https://arxiv.org/abs/2505.23664), [PDF](https://arxiv.org/pdf/2505.23664)\] \[Abstract: Bousso and Stanford (BS) argued that the black hole final state proposal leads to acausal effects and ill-defined probabilities for the [[0195 Firewall|AMPS]] experiment. We identify a loophole in their analysis using insights from [[0219 Entanglement wedge reconstruction|entanglement wedge reconstruction]] and replica wormholes. We trace the cause of the BS problems to the misidentification of the physical interior where the second AMPS measurement happens from among the multiple interiors introduced by the first measurement.\] # Antonini, Iliesiu, Rath, Tran ## A Black Hole Airy Tail \[Links: [arXiv](https://arxiv.org/abs/2507.10657), [PDF](https://arxiv.org/pdf/2507.10657)\] \[Abstract: In [[0050 JT gravity|Jackiw-Teitelboim (JT) gravity]], which is dual to a [[0197 Matrix model|random matrix ensemble]], the annealed entropy differs from the quenched entropy at low temperatures and goes negative. However, computing the quenched entropy in JT gravity requires a replica limit that is poorly understood. To circumvent this, we define an intermediate quantity called the semi-quenched entropy, which has the positivity properties of the quenched entropy, while requiring a much simpler replica trick. We compute this in JT gravity in different regimes using i) a bulk calculation involving wormholes corresponding to the Airy limit of the dual matrix integral and ii) a boundary calculation involving one-eigenvalue instantons, demonstrating consistency between these two calculations in their common regime of validity. We also clarify why a recent attempt to compute the quenched entropy using one-eigenvalue instantons is unreliable due to a breakdown of the saddle-point approximation for the one-eigenvalue instanton in the replica limit.\] # Arnaudo, Carballo, Withers ## QNM orthogonality relations for AdS black holes \[Links: [arXiv](https://arxiv.org/abs/2505.04696), [PDF](https://arxiv.org/pdf/2505.04696)\] \[Abstract: We present orthogonality relations for [[0325 Quasi-normal modes|quasinormal modes]] of a wide class of asymptotically AdS black holes. The definition is obtained from a standard product, modified by a CPT operator and placed on a complex radial contour which avoids branch points of the modes. They are inspired by existing constructions for de Sitter and Kerr spacetimes. The CPT operator is needed to map right eigenfunctions of the Hamiltonian into left eigenfunctions. The radial contour connects two copies of the dual QFT on a thermal [[0042 Schwinger-Keldysh techniques|Schwinger-Keldysh contour]], making contact with real-time holography and the double cone wormhole.\] # Baiguera, Balasubramanian, Caputa, Chapman, Haferkamp, Heller, Halpern (Review) ## Quantum complexity in gravity, quantum field theory, and quantum information science \[Links: [arXiv](https://arxiv.org/abs/2503.10753), [PDF](https://arxiv.org/pdf/2503.10753)\] \[Abstract: [[0204 Quantum complexity|Quantum complexity]] quantifies the difficulty of preparing a state or implementing a unitary transformation with limited resources. Applications range from quantum computation to condensed matter physics and quantum gravity. We seek to bridge the approaches of these fields, which define and study complexity using different frameworks and tools. We describe several definitions of complexity, along with their key properties. In quantum information theory, we focus on complexity growth in random quantum circuits. In quantum many-body systems and quantum field theory (QFT), we discuss a geometric definition of complexity in terms of geodesics on the unitary group. In dynamical systems, we explore a definition of complexity in terms of state or operator spreading, as well as concepts from [[0054 Tensor network|tensor-networks]]. We also outline applications to simple quantum systems, quantum many-body models, and QFTs including conformal field theories (CFTs). Finally, we explain the proposed relationship between complexity and gravitational observables within the holographic anti-de Sitter (AdS)/CFT correspondence.\] # Balasubramanian, Yildirim ## A Nonperturbative Toolkit for Quantum Gravity \[Links: [arXiv](https://arxiv.org/abs/2504.16986), [PDF](https://arxiv.org/pdf/2504.16986)\] \[Abstract: We propose a method for demonstrating equivalences beyond the saddlepoint approximation between quantities in quantum gravity that are defined by the [[0555 Gravitational path integral|Euclidean path integral]], without assumptions about [[0001 AdS-CFT|holographic duality]]. The method involves three ingredients: (1) a way of resolving the identity with an overcomplete basis of microstates that is under semiclassical control, (2) a drastic simplification of the sum over topologies in the limit where the basis is infinitely overcomplete, and (3) a way of cutting and splicing geometries to demonstrate equality between two different gravitational path integrals even if neither can be explicitly computed. We illustrate our methods by giving a general argument that the thermal partition function of quantum gravity with two boundaries factorises. One implication of our results is that universes containing a horizon can sometimes be understood as superpositions of horizonless geometries entangled with a [[0632 Closed universe|closed universe]].\] # Bao, Geng, Jiang ## Ryu-Takayanagi Formula for Multi-Boundary Black Holes from 2D Large-$c$ CFT Ensemble \[Links: [arXiv](https://arxiv.org/abs/2504.12388), [PDF](https://arxiv.org/pdf/2504.12388)\] \[Abstract: We study a class of quantum states involving multiple entangled CFTs in AdS${}_3$/CFT$_2$, associated with multi-boundary black hole geometries, and demonstrate that the [[0007 RT surface|Ryu-Takayanagi (RT) formula]] for [[0301 Entanglement entropy|entanglement entropy]] can be derived using only boundary CFT data. Approximating the OPE coefficients by their Gaussian moments within the 2D large-$c$ CFT ensemble, we show that both the norm of the states and the entanglement entropies associated with various bipartitions--reproducing the expected bulk dual results--can be computed purely from the CFT. All *macroscopic geometric* structures arising from gravitational saddles emerge entirely from the universal statistical moments of the *microscopic algebraic* CFT data, revealing a statistical-mechanical mechanism underlying semiclassical gravity. We establish a precise correspondence between the CFT norm, the [[0562 Liouville theory|Liouville]] partition function with [[0652 ZZ brane|ZZ boundary conditions]], and the exact gravitational path integral over 3D multi-boundary black hole geometries. For entanglement entropy, each RT phase arises from a distinct leading-order Gaussian contraction, with phase transitions--analogous to replica wormholes--emerging naturally from varying dominant statistical patterns in the CFT ensemble. Our derivation elucidates how the general mechanism behind holographic entropy, namely a boundary replica direction that elongates and becomes contractible in the bulk dual, is encoded explicitly in the statistical structure of the CFT data.\] # Barbon, Velasco-Aja ## A Note on Black Hole Entropy and Wormhole Instabilities \[Links: [arXiv](https://arxiv.org/abs/2502.00769), [PDF](https://arxiv.org/pdf/2502.00769)\] \[Abstract: We discuss recent approaches to the [[0248 Black hole microstates|computation]] of black hole entropies through semiclassical estimates of appropriate state overlaps, saturated by Euclidean wormhole configurations. We notice that the relevant saddle-point manifolds may exhibit instabilities, thereby compromising the interpretation of the Euclidean path integral as a tool for computing positive-definite inner products. We show that a proper treatment using a microcanonical formulation effectively addresses the puzzles posed by these instabilities.\] # Beetar, Graef, Murugan, Nastase, Van Zyl ## Krylov complexity, path integrals, and instantons \[Links: [arXiv](https://arxiv.org/abs/2507.13226), [PDF](https://arxiv.org/pdf/2507.13226)\] \[Abstract: [[0564 Krylov complexity|Krylov complexity]] has emerged as an important tool in the description of quantum information and, in particular, quantum chaos. Here we formulate Krylov complexity $K(t)$ for quantum mechanical systems as a path integral, and argue that at large times, for classical chaotic systems with at least two minima of the potential, that have a plateau for $K(t)$, the value of the plateau is described by quantum mechanical instantons, as is the case for standard transition amplitudes. We explain and test these ideas in a simple toy model.\] # Bhattacharya, Padhi, Sharma, Singha ## Thermal Product Formula for Shear Modes \[Links: [arXiv](https://arxiv.org/abs/2504.17781), [PDF](https://arxiv.org/pdf/2504.17781)\] \[Abstract: We investigate the validity of the [[0664 Thermal product formula|thermal product formula]] proposed in [[2023#Dodelson, Iossa, Karlsson, Zhiboedov]], for the shear channel fluctuations of R-charged black branes in AdS$_5$ where the shear mode is coupled with charge diffusion mode at non-zero momentum. When these modes are suitably decoupled, we are able to obtain an exact formula for the two point functions of the boundary current and energy-momentum tensor in terms of the [[0325 Quasi-normal modes|quasinormal modes]] of this channel. This exact formula is a simple modification of the previous version of the product formula. We also obtain a similar formula for the case involving a boundary global R-symmetry anomaly, when we have a bulk [[0089 Chern-Simons theory|Chern-Simons]] term which introduces additional couplings in the shear channel. Also based on insights from the quasinormal mode spectrum, we report on an instability as well as the presence of high momentum long-lived modes associated with large values of the anomaly coefficient.\] # Bianchi, Mattiello, Sisti ## The entropy of radiation for local quenches in higher dimensions \[Links: [arXiv](https://arxiv.org/abs/2502.00105), [PDF](https://arxiv.org/pdf/2502.00105)\] \[Abstract: We investigate the real time dynamics of the radiation produced by a local [[0558 Quantum quench|quench]] in a $d$-dimensional conformal field theory (CFT) with $d>2$. Using the interpretation of the higher-dimensional twist operator as a conformal defect, we study the time evolution of the [[0301 Entanglement entropy|entanglement entropy]] of the radiation across a spherical entangling surface. We provide an analytic estimate for the early- and late-time behavior of the entanglement entropy and derive an upper bound valid at all times. We extend our analysis to the case of a [[0548 Boundary CFT|boundary CFT]] (BCFT) and derive similar results through a detailed discussion of the setup with two conformal defects (the boundary and the twist operator). We conclude with a holographic analysis of the process, computing the time evolution of the [[0007 RT surface|holographic entanglement entropy]] (HEE) as the area of the [[0007 RT surface|Ryu-Takayanagi surface]] in a backreacted geometry. This gives a Page-like curve in agreement with the early- and late-time results obtained with CFT methods. The extension to a [[0181 AdS-BCFT|holographic BCFT]] setup is generically hard and we consider the case of a tensionless end-of-the-world brane.\] # Blacker, Castro, Sybesma, Toldo ## Quantum corrections to the path integral of near extremal de Sitter black holes \[Links: [arXiv](https://arxiv.org/abs/2503.14623), [PDF](https://arxiv.org/pdf/2503.14623)\] \[Abstract: We study quantum corrections to the Euclidean path integral of charged and static four-dimensional de Sitter (dS$_4$) black holes near extremality. These black holes admit three different extremal limits (Cold, Nariai and Ultracold) which exhibit $AdS_2 \times S^2$, $dS_2$ $\times S^2$ and $\text{Mink}_2 \times S^2$ near horizon geometries, respectively. The one-loop correction to the [[0555 Gravitational path integral|gravitational path integral]] in the near horizon geometry is plagued by infrared divergencies due to the presence of tensor, vector and gauge zero modes. Inspired by the analysis of black holes in flat space, we regulate these divergences by introducing a small temperature correction in the Cold and Nariai background geometries. In the Cold case, we find a contribution from the gauge modes which is not present in previous work in asymptotically flat spacetimes. Several issues concerning the Nariai case, including the presence of negative norm states and negative eigenvalues, are discussed, together with problems faced when trying to apply this procedure to the Ultracold solution.\] # Blommaert, Kudler-Flam, Urbach ## Absolute entropy and the observer's no-boundary state \[Links: [arXiv](https://arxiv.org/abs/2505.14771), [PDF](https://arxiv.org/pdf/2505.14771)\] \[Abstract: We investigate the [[0162 No-boundary wavefunction|no-boundary proposal]] for [[0632 Closed universe|closed universes]] with an observer. We argue that the observer's no-boundary state is the identity operator on the physical Hilbert space, i.e. the maximum entropy state. Geometrically, the no-boundary state is a [[0216 Bra-ket wormholes|bra-ket wormhole]]. Our result is consistent with all previously discussed cases of traces for invariantly defined regions. We explicitly show that the observer's no-boundary state is an identity operator in [[0050 JT gravity|Jackiw-Teitelboim gravity]]. Expectation values in the no-boundary state provide a trace for the observer's algebra, which allows one to define [[0301 Entanglement entropy|von Neumann entropy]] for observers in different universes as the relative entropy with respect to the no-boundary state.\] # Blommaert, Levine ## Sphere amplitudes and observing the universe's size \[Links: [arXiv](https://arxiv.org/abs/2505.24633), [PDF](https://arxiv.org/pdf/2505.24633)\] \[Abstract: Sine dilaton gravity is holographically related to [[0503 Double-scaled SYK|DSSYK]]. We explain how to interpret sine dilaton as 2d quantum cosmology. This paves the way for using two copies of DSSYK as hologram for Big-Bang cosmologies. We study the most basic cosmological observable: the sphere amplitude. Via canonical quantization we find a finite answer that matches the on-shell action of a dual matrix integral. The sphere amplitude (or the norm of the [[0162 No-boundary wavefunction|no-boundary wavefunction]]) also gives a prediction for the universe's size. In the context of slow-roll inflation, the no-boundary state is non-normalizable, and predicts a small universe, in contradiction with experiments. We argue that an avatar of these issues exists in dS [[0050 JT gravity|JT gravity]]. By considering sine dilaton as a UV completion of dS JT gravity, the state becomes normalizable. We then consider the observer's no-boundary state and show that this prefers neither small nor large universes. The resulting distribution is flat.\] # Blommaert, Levine, Mertens, Papalini, Parmentier ## Wormholes, branes and finite matrices in sine dilaton gravity \[Links: [arXiv](https://arxiv.org/abs/2501.17091), [PDF](https://arxiv.org/pdf/2501.17091)\] \[Abstract: We compute the double trumpet in sine dilaton gravity via WdW quantization. The wormhole size is discretized. The wormhole amplitude matches the spectral correlation of a finite-cut matrix integral, where matrices have large but finite dimensions. This strongly suggests an identification of the sine dilaton gravity theory with the $q$-deformed [[0050 JT gravity|JT gravity]] matrix integral. At the very least, it captures all universal content of that matrix model. The disk decomposes into the physical (gauge invariant) solutions of the WdW equation, which are trumpets with discrete sizes. This decomposition modifies the usual no-boundary wavefunction to a normalizable one in sine dilaton gravity. We furthermore present an exact quantization of sine dilaton gravity with open and closed end of the world branes. These EOW branes correspond with [[0658 FZZT brane|FZZT branes]] for the two [[0562 Liouville theory|Liouville theories]] that make up sine dilaton gravity. The WdW equation implies redundancies in this space of branes, leaving a one parameter family of gauge invariant branes. One gauge choice corresponds with branes discussed by Okuyama in the context of chord diagrams and of [[0503 Double-scaled SYK|DSSYK]]. Legendre transforming the EOW brane amplitude reproduces the trumpet, independent of the WdW quantization calculation. One could read our work as fleshing out the Hilbert space of [[0632 Closed universe|closed universes]] in sine dilaton gravity.\] # Bolokhov, Skvortsova (Review) ## Review of analytic results on quasinormal modes of black holes \[Links: [arXiv](https://arxiv.org/abs/2504.05014), [PDF](https://arxiv.org/pdf/2504.05014)\] \[Abstract: We present a concise review of known analytic results for [[0325 Quasi-normal modes|quasinormal modes]] of black holes and related spacetimes. Our emphasis is on those regimes where the perturbation equations admit exact or perturbative solutions, providing insights complementary to numerical or semi-analytic approaches. We discuss solvable cases in lower-dimensional spacetimes, algebraically special modes, and exact results in [[0006 Higher-derivative gravity|higher-curvature gravity]] theories. Particular attention is given to the eikonal regime and its correspondence with null geodesics, as well as to beyond-eikonal approximations based on inverse multipole expansions in parametrized metrics. We review analytic solutions obtained in the near-extremal limit of Schwarzschild-de Sitter black holes, in the regime of large field mass, and in pure de Sitter and anti-de Sitter spacetimes, where boundary conditions play a crucial role. While not exhaustive, this overview highlights the diversity of techniques and physical insights made possible by analytic treatments of quasinormal spectra.\] # Boruch, Di Ubaldo, Haehl, Perlmutter, Rozali ## Modular-invariant random matrix theory and AdS${}_3$ wormholes \[Links: [arXiv](https://arxiv.org/abs/2503.00101), [PDF](https://arxiv.org/pdf/2503.00101)\] \[Abstract: We develop a non-perturbative definition of RMT${}_2$: a generalization of [[0579 Random matrix theory|random matrix theory]] that is compatible with the symmetries of two-dimensional conformal field theory. Given any random matrix ensemble, its $n$-point spectral correlations admit a prescribed modular-invariant lift to RMT${}_2$, which moreover reduce to the original random matrix correlators in a near-extremal limit. Central to the prescription is a presentation of random matrix theory in Mellin space, which lifts to two dimensions via the $\text{SL}(2,\mathbb{Z})$ spectral decomposition employed in previous work. As a demonstration we perform the explicit RMT${}_2$ lift of two-point correlations of the GUE Airy model. We propose that in AdS$_3$ pure gravity, semiclassical amplitudes for off-shell $n$-boundary torus wormholes with topology $\Sigma_{0,n} \times S^1$ are given by the RMT${}_2$ lift of [[0050 JT gravity|JT gravity]] wormhole amplitudes. For the three-boundary case, we identify a gravity calculation which matches the RMT${}_2$ result.\] # Bourne, Fliss, Knighton ## A spool for every quotient: One-loop partition functions in AdS$_3$ gravity \[Links: [arXiv](https://arxiv.org/abs/2507.05364), [PDF](https://arxiv.org/pdf/2507.05364)\] \[Abstract: The [[0653 Wilson spool|Wilson spool]] is a prescription for expressing one-loop determinants as topological line operators in [[0002 3D gravity|three-dimensional gravity]]. We extend this program to describe massive spinning fields on all smooth, cusp-free, solutions of Euclidean gravity with a negative cosmological constant. Our prescription makes use of the expression of such solutions as a quotients of hyperbolic space. The result is a gauge-invariant topological operator, which can be promoted to an off-shell operator in the gravitational path integral about a given saddle-point. When evaluated on-shell, the Wilson spool reproduces and extends the known results of one-loop determinants on hyperbolic quotients. We motivate our construction of the Wilson spool from multiple perspectives: the [[0630 Selberg trace formula|Selberg trace formula]], worldline quantum mechanics, and the [[0325 Quasi-normal modes|quasinormal mode]] method.\] # Bousso, Kaya ## Holographic Entropy Cone Beyond AdS/CFT \[Links: [arXiv](https://arxiv.org/abs/2502.03516), [PDF](https://arxiv.org/pdf/2502.03516)\] \[Abstract: We extend all known area inequalities obeyed by [[0007 RT surface|Ryu-Takayanagi surfaces]] of AdS boundary regions -- the [[0259 Holographic entropy cone|holographic entropy cone]] -- to static generalized entanglement wedges of bulk regions in arbitrary spacetimes. The generalized holographic entropy cone is subject to a mutual independence condition on the bulk regions: each bulk input region must be outside the entanglement wedge of the union of all others. The condition captures when gravitating regions involve fundamentally distinct degrees of freedom despite the nonlocality inherent in the holographic principle.\] # Buric, Gusev, Parnachev ## Thermal holographic correlators and KMS condition \[Links: [arXiv](https://arxiv.org/abs/2505.10277), [PDF](https://arxiv.org/pdf/2505.10277)\] \[Abstract: Thermal [[0103 Two-point functions|two-point functions]] in holographic CFTs receive contributions from two parts. One part comes from the identity, the stress tensor and multi-stress tensors and constitutes the stress-tensor sector. The other part consists of contributions from double-trace operators. The sum of these two parts must satisfy the [[0521 KMS condition|KMS condition]] -- it has to be periodic in Euclidean time. The stress-tensor sector can be computed by analyzing the bulk equations of motions near the AdS boundary and is not periodic by itself. We show that starting from the expression for the stress-tensor sector one can impose the KMS condition to fix the double-trace part, and hence the whole correlator. We perform explicit calculations in the asymptotic approximation, where the stress-tensor sector can be computed exactly. One can either sum over the thermal images of the stress-tensor sector and subtract the singularities or solve for the KMS condition directly and perform the Borel resummation of the resulting double-trace data -- the results are the same.\] # Caminiti, Capeccia, Ciambelli, Myers ## Geometric modular flows in 2d CFT and beyond \[Links: [arXiv](https://arxiv.org/abs/2502.02633), [PDF](https://arxiv.org/pdf/2502.02633)\] \[Abstract: We study geometric modular flows in [[0003 2D CFT|two-dimensional conformal field theories]]. We explore which states exhibit a geometric [[0416 Modular Hamiltonian|modular flow]] with respect to a causally complete subregion and, conversely, how to construct a state from a given geometric modular flow. Given suitable boundary conditions, we find that generic geometric modular flows in the Rindler wedge are conformally equivalent. Based on this insight, we show how conformal unitaries can be used to explicitly construct a state for each flow. We analyze these states, deriving general formulas for the energy density and [[0301 Entanglement entropy|entanglement entropy]]. We also consider geometric flows beyond the Rindler wedge setting, and in higher dimensions.\] # Caron-Huot, Chakravarty, Namjou ## Boundary imprint of bulk causality \[Links: [arXiv](https://arxiv.org/abs/2501.13182), [PDF](https://arxiv.org/pdf/2501.13182)\] \[Abstract: Motivated by the holographic correspondence, we study the boundary imprint of bulk lightcones in spacetimes with boundaries. These lightcones can be observed whenever a localized event takes place in the bulk. The associated boundary surfaces (hyperboloids) [[0027 Bulk reconstruction using lightcone cuts|reveal the bulk conformal metric]]. We work out a Hamilton-Jacobi description of these surfaces and analyze them in explicit examples. Bulk causality translates into a boundary inclusion property from which the bulk geodesic equation can be derived under some assumptions.\] # Chakravarty, Maloney, Namjou, Ross ## The spectrum of pure dS$_3$ gravity in the static patch \[Links: [arXiv](https://arxiv.org/abs/2505.06420), [PDF](https://arxiv.org/pdf/2505.06420)\] \[Abstract: We consider the quantum mechanical description of the de Sitter static patch in three-dimensional general relativity. We consider a [[0464 Lorentzian path integral|Lorentzian path integral]] that conjecturally computes the Fourier transform of the spectrum of the static patch Hamiltonian. We regulate a saddle point for this integral by a complex deformation that connects it to future infinity. Our computation is thus closely connected with the wave function of [[0545 de Sitter quantum gravity|de Sitter gravity]] on a torus at future infinity. Motivated by this, we identify an infinite number of saddle points that contribute to our Lorentzian path integral. Their sum gives a surprisingly simple result, which agrees with the expected features of the de Sitter static patch. For example, the thermal entropy, evaluated at the de Sitter temperature, agrees with the Bekenstein-Hawking formula. We also obtain a spectrum in the spin-zero sector, which is bounded, discrete, and has an integer degeneracy of states. It includes a dense spectrum of states, making both positive and negative contributions to the trace, arranged in such a way that negative contributions are invisible in the computation of any smooth observable. Nevertheless, several mysteries remain.\] # Chen ## Observers seeing gravitational Hilbert spaces: abstract sources for an abstract path integral \[Links: [arXiv](https://arxiv.org/abs/2505.15892), [PDF](https://arxiv.org/pdf/2505.15892)\] \[Abstract: The [[0555 Gravitational path integral|gravitational path integral]] suggests a striking result: the Hilbert space of [[0632 Closed universe|closed universes]] in each superselection sector, a so-called $\alpha$-sector, is one-dimensional. We develop an abstract formalism encapsulating recent proposals that modify the gravitational path integral in the presence of observers and allow larger Hilbert spaces to be associated with closed universes. Our formalism regards the gravitational path integral as a map from abstract objects called sources to complex numbers, and introduces additional objects called partial sources, which form sources when glued together. We apply this formalism to treat, on equal footing, universes with spatial boundaries, closed universes with prescribed observer worldlines, and closed universes containing observers entangled with external systems. In these contexts, the relevant gravitational Hilbert spaces contain states prepared by partial sources and can consequently have nontrivial $\alpha$-sectors supporting noncommuting operators. Within our general framework, the positivity of the gravitational inner product implies a bound on the Hilbert space trace of certain positive operators over each $\alpha$-sector. The trace of such operators, in turn, quantifies the effective size of this Hilbert space.\] # Chua, Hartman, Weng ## Replica manifolds, pole skipping, and the butterfly effect \[Links: [arXiv](https://arxiv.org/abs/2504.08139), [PDF](https://arxiv.org/pdf/2504.08139)\] \[Abstract: The black hole butterfly effect is a signal of quantum chaos in holographic theories that can be probed in different ways, including [[0482 Out-of-time-order correlator|out-of-time-order correlators]] (OTOCs), [[0179 Pole skipping|pole skipping]] (PS), and [[0219 Entanglement wedge reconstruction|entanglement wedge (EW) reconstruction]]. Each of these three phenomena can be used to define a [[0167 Butterfly velocity|butterfly velocity]] that measures the speed at which chaos spreads. In a general quantum system the three velocities $v_B^{\text{OTOC}}$, $v_B^{\text{PS}}$, and $v_B^{\text{EW}}$ can be different, but it is known from explicit calculations that they are all equal in certain holographic theories dual to Einstein gravity plus [[0006 Higher-derivative gravity|higher-curvature corrections]]. A conceptual explanation for this apparent coincidence is lacking. We show that it follows from a deeper relationship: The pole-skipping mode, added to the black hole background, can be reinterpreted as the gravitational replica manifold for the late-time entanglement wedge, and its imaginary part is the shockwave that computes the OTOC. Thus pole skipping is directly related to [[0522 Entanglement dynamics|entanglement dynamics]] in holographic theories, and the origin of the pole-skipping mode is an extremal surface on the horizon. This explains the coincidence $v_B^{\text{OTOC}} = v_B^{\text{PS}} = v_B^{\text{EW}}$ in known cases, and extends it to general theories of gravity with a pole-skipping mode having the usual behavior.\] # Colin-Ellerin, Lin, Penington ## Generalized entropy of gravitational fluctuations \[Links: [arXiv](https://arxiv.org/abs/2501.08308), [PDF](https://arxiv.org/pdf/2501.08308)\] \[Abstract: The corrections to [[0007 RT surface|holographic entanglement entropy]] from bulk quantum fields in a classical gravitational background are now well understood. They lead, in particular, to unitary Page curves for evaporating black holes. However, the correct treatment of quantum fluctuations of the metric, including graviton excitations, is a longstanding problem. We provide a gauge-invariant prescription for the generalized entropy of gravitons in anti-de Sitter space in terms of areas and bulk entanglement entropy, generalizing the [[0212 Quantum extremal surface|quantum extremal surface]] prescription to accommodate fluctuations in the semiclassical spacetime geometry. This task requires a careful treatment of the area operator on the graviton Hilbert space and the definition of a "quantum extremal gauge" in which the extremal surface is unperturbed. It also requires us to determine the correct vacuum modular Hamiltonian for the graviton field, which we fix by requiring that it doesn't contain a boundary term in extremal gauge. We check our prescription with an explicit computation of the vacuum-subtracted generalized entropy of states containing a graviton in an AdS-Rindler background. Our results exactly match vacuum-subtracted [[0301 Entanglement entropy|von Neumann entropies]] for stress-tensor excited states in holographic conformal field theory with $d>2$ dimensions. We also use covariant phase space techniques to give a partial proof of our prescription when the entanglement wedge for the background spacetime has a bifurcate Killing horizon. Along the way, we identify a class of perturbative graviton states that have parametrically larger generalized entropy, in the small $G_N$ expansion, than any low-energy excitations of an ordinary quantum field.\] # Collier, Eberhardt, Muhlmann (a) ## A microscopic realization of dS$_3$ \[Links: [arXiv](https://arxiv.org/abs/2501.01486), [PDF](https://arxiv.org/pdf/2501.01486)\] \[Abstract: We propose a precise duality between pure de Sitter quantum gravity in 2+1 dimensions and a double-scaled matrix integral. This duality unfolds in two distinct aspects. First, by carefully quantizing the gravitational phase space, we arrive at a novel proposal for the quantum state of the universe at future infinity. We compute cosmological correlators of massive particles in the universe specified by this wavefunction. Integrating these correlators over the metric at future infinity yields gauge-invariant observables, which are identified with the string amplitudes of the [[0650 Complex Liouville string|complex Liouville string]] [arXiv:2409.17246](https://arxiv.org/abs/2409.17246). This establishes a direct connection between integrated cosmological correlators and the resolvents of the matrix integral dual to the complex Liouville string, thereby demonstrating one aspect of the dS$_3$/matrix integral duality. The second aspect concerns the cosmological horizon of the dS static patch and the Gibbons-Hawking entropy it is conjectured to encode. We show that this entropy can be reproduced exactly by counting the entries of the matrix.\] # Collier, Eberhardt, Muhlmann (b) ## The complex Liouville string: the gravitational path integral \[Links: [arXiv](https://arxiv.org/abs/2501.10265), [PDF](https://arxiv.org/pdf/2501.10265)\] \[Abstract: We give a rigorous definition of sine dilaton gravity in terms of the worldsheet theory of the [[0650 Complex Liouville string|complex Liouville string]] [arXiv:2409.17246](https://arxiv.org/abs/2409.17246). The latter has a known exact solution that we leverage to explore the gravitational path integral of sine dilaton gravity - a quantum deformation of dS [[0050 JT gravity|JT gravity]] that admits both AdS$_2$ and dS$_2$ vacua. We uncover that the gravitational path integral receives contributions from new saddles describing transitions between vacua in a third-quantized picture. We also discuss the sphere and disk partition function in this context and contrast our findings with other recent work on this theory.\] # Czech, Shuai, Wang ## Entropy Inequalities Constrain Holographic Erasure Correction \[Links: [arXiv](https://arxiv.org/abs/2502.12246), [PDF](https://arxiv.org/pdf/2502.12246)\] \[Abstract: We interpret [[0259 Holographic entropy cone|holographic entropy inequalities]] in terms of erasure correction. The non-saturation of an inequality is a necessary condition for certain schemes of holographic erasure correction, manifested in the bulk as non-empty overlaps of corresponding entanglement wedges.\] # de Boer, Kames-King, Post ## Surgery and statistics in 3d gravity \[Links: [arXiv](https://arxiv.org/abs/2506.04151), [PDF](https://arxiv.org/pdf/2506.04151)\] \[Abstract: We extend the correspondence between [[0663 OPE statistics|universal statistical]] features of large-$c$ 2d CFTs and surgery methods in pure AdS$_3$ quantum gravity. In particular, we introduce a method that we call [[0579 Random matrix theory|RMT]] surgery, which relates a large class of off-shell partition functions in 3d gravity to the spectral statistics of general CFT observables. We apply this method to construct and compute an off-shell Euclidean wormhole whose boundaries are four-punctured spheres, which captures level repulsion in the high-energy sector of the boundary CFT. Using a similar gluing prescription, we also explore a new class of off-shell torus wormholes with trumpet boundaries, contributing to statistical moments of the density of primary states. Lastly, we demonstrate that surgery methods can be used as an intermediate step towards computing Seifert manifolds directly in [[0002 3D gravity|3d gravity]].\] # Dey, Nanda, Roy, Sake, Trivedi ## JT Gravity in de Sitter Space and Its Extensions \[Links: [arXiv](https://arxiv.org/abs/2501.03148), [PDF](https://arxiv.org/pdf/2501.03148)\] \[Abstract: We discuss and extend some aspects pertaining to the canonical quantisation of [[0050 JT gravity|JT gravity]] in [[0545 de Sitter quantum gravity|de Sitter]] space, including the problem of time and the construction of a Hilbert space. We then extend this discussion to other two dimensional models obtained by changing the dilaton potential and show that the canonical quantisation procedure can be carried out for a large class of such models. Some discussion leading towards a [[0555 Gravitational path integral|path integral]] understanding for states, other than the [[0162 No-boundary wavefunction|Hartle Hawking state]], is also included here, along with comments pertaining to Holography and the entropy of de Sitter space.\] # Dodelson ## Black holes from chaos \[Links: [arXiv](https://arxiv.org/abs/2501.06170), [PDF](https://arxiv.org/pdf/2501.06170)\] \[Abstract: We study the emergence of black hole geometry from chaotic systems at finite temperature. The essential input is the universal operator growth hypothesis, which dictates the asymptotic behavior of the Lanczos coefficients. Under this assumption, we map the chaotic dynamics to a discrete analog of the scattering problem on a black hole background. We give a simple prescription for computing the Green's functions, and explore some of the resulting analytic properties. In particular, assuming that the Lanczos coefficients are sufficiently smooth, we present evidence that the spectral density is a meromorphic function of frequency with no zeroes. Our formalism provides a framework for accurately computing the late time behavior of Green's functions in chaotic systems, and we work out several instructive examples.\] # Dong, Marolf, Rath ## Geometric Entropies and their Hamiltonian Flows \[Links: [arXiv](https://arxiv.org/abs/2501.12438), [PDF](https://arxiv.org/pdf/2501.12438)\] \[Abstract: In holographic theories, the [[0007 RT surface|Hubeny-Rangamani-Takayanagi]] (HRT) area operator plays a key role in our understanding of the emergence of semiclassical Einstein-Hilbert gravity. When [[0006 Higher-derivative gravity|higher derivative corrections]] are included, the role of the area is instead played by a more general functional known as the geometric entropy. It is thus of interest to understand the flow generated by the geometric entropy on the classical phase space. In particular, the fact that the associated flow in Einstein-Hilbert or [[0050 JT gravity|Jackiw-Teitelboim (JT) gravity]] induces a relative boost between the left and right entanglement wedges is deeply related to the fact that gravitational dressing promotes the [[0415 Von Neumann algebra|von Neumann algebra]] of local fields in each wedge to type II. This relative boost is known as a [[0483 Boundary-condition-preserving kink transformation|boundary-condition-preserving (BCP) kink-transformation]]. In a general theory of gravity (with arbitrary higher-derivative terms), it is straightforward to show that the flow continues to take the above geometric form when acting on a spacetime where the HRT surface is the bifurcation surface of a Killing horizon. However, the form of the flow on other spacetimes is less clear. In this paper, we use the manifestly-covariant [[0150 Peierls bracket|Peierls bracket]] to explore such flows in two-dimensional theories of JT gravity coupled to matter fields with higher derivative interactions. The results no longer take a purely geometric form and, instead, demonstrate new features that should be expected of such flows in general higher derivative theories. We also show how to obtain the above flows using [[0360 Poisson bracket|Poisson brackets]].\] # Engelhardt, Gesteau ## Further Evidence Against a Semiclassical Baby Universe in AdS/CFT \[Links: [arXiv](https://arxiv.org/abs/2504.14586), [PDF](https://arxiv.org/pdf/2504.14586)\] \[Abstract: We argue that a large class of asymptotically AdS geometries with a semiclassical [[0051 Baby universes|baby universe]] cannot be realized within the AdS/CFT correspondence. This in particular resolves a recent puzzle introduced by Antonini and Rath, in which a single CFT state appeared to simultaneously describe an AdS spacetime with a baby universe and one without. We construct a low-energy (and low complexity) boundary operator whose expectation values in the descriptions with and without the baby universe cannot match if the baby universe is semiclassical. This operator conclusively identifies the actual bulk dual: the spacetime without a semiclassical baby universe. This result assumes only that AdS/CFT admits an extrapolate dictionary and an asymptotically isometric encoding of the causal wedge into the dual CFT, without which the correspondence may well be vacuous.\] # Engelhardt, Gesteau, Harlow ## Observer complementarity for black holes and holography \[Links: [arXiv](https://arxiv.org/abs/2507.06046), [PDF](https://arxiv.org/pdf/2507.06046)\] \[Abstract: We present a mathematical formulation of [[0347 Black hole complementarity|black hole complementarity]] based on recent rules for including the observer in quantum cosmology. We argue that this provides a self-consistent treatment of the interior of an evaporating black hole throughout its history, as well as the Antonini-Sasieta-Swingle-Rath configuration where a [[0632 Closed universe|closed universe]] is entangled with a pair of AdS universes.\] # Estienne, Lin ## Entanglement Entropy and Cauchy-Hadamard Renormalization \[Links: [arXiv](https://arxiv.org/abs/2501.19014), [PDF](https://arxiv.org/pdf/2501.19014)\] \[Abstract: This note presents a purely geometric construction of the so-called twist-field correlation functions in Conformal Field Theory (CFT), derived from conical singularities. This approach provides a purely mathematical interpretation of the seminal results in physics by Cardy and Calabrese on the [[0301 Entanglement entropy|entanglement entropy]] of quantum systems. Specifically, we begin by defining CFT partition functions on surfaces with conical singularities, using a ''Cauchy-Hadamard renormalization'' of the Polyakov anomaly integral. Next, we demonstrate that for a branched cover $f:\Sigma_d\to \Sigma$ with $d$ sheets, where the cover inherits the pullback of a smooth metric from the base, a specific ratio of partition functions on the cover to the base transforms under conformal changes of the base metric in the same way as a correlation function of CFT primary fields with specific conformal weights. We also provide a discussion of the physical background and motivation for entanglement entropy, focusing on path integrals and the replica trick, which serves as an introduction to these ideas for a mathematical audience.\] # Fliss ## Massive fields and Wilson spools in JT gravity \[Links: [arXiv](https://arxiv.org/abs/2503.08657), [PDF](https://arxiv.org/pdf/2503.08657)\] \[Abstract: We give a prescription for minimally coupling massive matter to [[0050 JT gravity|JT gravity]] with either sign of cosmological constant directly in its formulation as a topological [[0557 BF theory|BF theory]]. This coupling takes the form of a '[[0653 Wilson spool|Wilson spool]],' originally introduced in the context of [[0002 3D gravity|three-dimensional gravity]]. The Wilson spool expresses the exact one-loop partition function as the integral over a Wilson loop operator. We construct the spool by considering the partition function of a massive scalar field on Euclidean dS$_2$ and on Euclidean AdS$_2$. We discuss its extension to other geometries (including the 'trumpet' and conical defect geometries) and its relation to the three-dimensional spool through dimensional reduction.\] # Folkestad ## Holographic Time Crystals vs Penrose \[Links: [arXiv](https://arxiv.org/abs/2502.01723), [PDF](https://arxiv.org/pdf/2502.01723)\] \[Abstract: In the large-$N$ limit, no known no-go theorem rules out thermal time crystals that spontaneously break continuous time-translation, unlike in the large volume limit. If thermal time crystals exist in holographic CFTs, they would correspond to ensemble-dominating black holes with eternally time-varying exterior geometries. We point out that recent work on a conjectured non-linear instability of slowly rotating Kerr-AdS$_4$ produced viable candidates for such states. Then we show that the existence of holographic microcanonical time crystals would imply violations of the AdS [[0476 Penrose inequality|Penrose inequality]] (PI). We proceed to look for violations of the PI in spherical symmetry, working with Einstein-scalar gravity with the most general possible boundary conditions compatible with boundary conformal invariance. We derive a set of ODEs for maximally PI-violating initial data. Solving these numerically, we find strong evidence that in the particular case of spherical symmetry, the PI holds iff the [[0116 Positive energy theorem|positive mass theorem]] (PMT) holds. This suggests that holographic CFT$_3$ time crystals can only possibly exist at non-zero angular momentum, at least in the absence of electric charge. We also discover neutral hairy black holes in a consistent truncation of M-theory that has a PMT and boundary conditions respecting conformal invariance, disproving an existing no-hair conjecture. Finally, we show that previous PI-violating solutions by the author all existed in theories where the PMT is violated. Unfortunately, our results imply that there currently are no known examples where the PI functions as a non-trivial Swampland constraint.\] # Fu, Izumi, Yoshida ## Consistency between Bulk and Boundary Causalities in Asymptotically Anti-de Sitter Spacetimes \[Links: [arXiv](https://arxiv.org/abs/2504.15910), [PDF](https://arxiv.org/pdf/2504.15910)\] \[Abstract: We investigate the consistency between bulk and [[0091 Boundary causality|boundary causalities]] in static, spherically symmetric, asymptotically anti-de Sitter (AdS) spacetimes. We derive a general formula that provides sufficient conditions for time advance, where bulk propagation arrives earlier than any boundary propagation. As an application, we show that in Reissner--Nordström--anti de Sitter spacetime, no geodesic satisfies the sufficient conditions for time advance even in the super-extremal case. Furthermore, we demonstrate that the Einstein--Euler--Heisenberg theory exhibits time advance when one or a linear combination of the coupling constants is positive and below an upper bound determined by the AdS length scale.\] # Furugori, Nishii, Yoshida, Yoshimura ## Apparent Horizons Associated with Dynamical Black Hole Entropy \[Links: [arXiv](https://arxiv.org/abs/2507.14105), [PDF](https://arxiv.org/pdf/2507.14105)\] \[Abstract: We define entropic [[0298 MOTS|marginally outer trapped surfaces]] (E-MOTSs) as a generalization of apparent horizons. We then show that, under first-order perturbations around a stationary black hole, the dynamical [[0004 Black hole entropy|black hole entropy]] proposed by Hollands, Wald, and Zhang, defined on a background Killing horizon, can be expressed as the Wall entropy evaluated on an E-MOTS associated with it. Our result ensures that the Hollands-Wald-Zhang entropy reduces to the standard [[0559 Wald entropy|Wald entropy]] in each stationary regime of a dynamical black hole, thereby reinforcing the robustness of the dynamical entropy formulation.\] # Geng, Hung, Jiang ## It from ETH: Multi-interval Entanglement and Replica Wormholes from Large-$c$ BCFT Ensemble \[Links: [arXiv](https://arxiv.org/abs/2505.20385), [PDF](https://arxiv.org/pdf/2505.20385)\] \[Abstract: We provide a derivation of the [[0007 RT surface|Ryu-Takayanagi (RT) formula]] in [[0002 3D gravity|3D gravity]] for generic boundary subsystems--including RT surface phase transitions--directly from the dual [[0003 2D CFT|two-dimensional conformal field theory]] (CFT). Our approach relies on the [[0663 OPE statistics|universal statistics]] of the algebraic conformal data and the large-$c$ behavior of conformal blocks with Cardy boundaries involved. We observe the emergence of 3D multi-boundary black holes with Karch-Randall branes from entangled states of any number of CFT's with and without Cardy boundaries. We obtain the RT formula from the CFT, in the high-temperature regime. Two direct applications are: $\textbf{1)}$ A simple derivation of the multi-interval entanglement entropy for the vacuum state of a single CFT; $\textbf{2)}$ A CFT-based detection of the emergence of replica wormholes in the context of entanglement [[0213 Islands|islands]] and black hole [[0248 Black hole microstates|microstate counting]]. Our framework yields the first holographic random [[0054 Tensor network|tensor network]] that faithfully captures the entanglement structure of holographic CFTs. These results imply that bulk spacetime geometries indeed emerge from the [[0040 Eigenstate thermalisation hypothesis|eigenstate thermalization hypothesis]] (ETH) in the dual field theory in the large-$c$ limit--a paradigm we refer to as *It from ETH*.\] # Giotopoulos (Notes) ## Covariant Lie Derivatives and (Super-)Gravity \[Links: [arXiv](https://arxiv.org/abs/2507.00140), [PDF](https://arxiv.org/pdf/2507.00140)\] \[Abstract: The slightly subtle notion of covariant Lie derivatives of *bundle-valued* differential forms is crucial in many applications in physics, notably in the computation of conserved currents in gauge theories, and yet the literature on the topic has remained fragmentary. This note provides a complete and concise mathematical account of covariant Lie derivatives on a spacetime (super-)manifold M, defined via choices of lifts of spacetime vector fields to principal $G$-bundles over it, or equivalently, choices of covariantization correction terms on spacetime. As an application in the context of (super-)gravity, two important examples of covariant Lie derivatives are presented in detail, which have not appeared in unison and direct comparison: **(i)** The natural covariant Lie derivative relating (super-)diffeomorphism invariance to local translational (super-)symmetry, and **(ii)** the [[0527 Lie derivative of spinor fields|Kosmann Lie derivative]] relevant to the description of isometries of (super-)gravity backgrounds. Finally, we use the latter to rigorously justify the usage of the traditional (non-covariant) Lie derivative on coframes and associated fields in dimensional reduction scenarios along abelian $G$-fibers, an issue which has thus far remained open for topologically non-trivial spacetimes.\] # Harlow, Usatyuk, Zhao ## Quantum mechanics and observers for gravity in a closed universe \[Links: [arXiv](https://arxiv.org/abs/2501.02359), [PDF](https://arxiv.org/pdf/2501.02359)\] \[Abstract: Recent arguments based on the [[0212 Quantum extremal surface|quantum extremal surface]] formula or the gravitational path integral have given fairly compelling evidence that the Hilbert space of quantum gravity in a [[0632 Closed universe|closed universe]] is one-dimensional and real. How can this be consistent with the complexity of our own experiences? In this paper we propose that the experiences of any observer $Ob$ in a closed universe can be approximately described by a quantum mechanical theory with a Hilbert space whose dimension is roughly $e^{S_{Ob}}$, where $S_{Ob}$ is the number of degrees of freedom of $Ob$. Moreover we argue that the errors in this description are exponentially small in $S_{Ob}$. We give evidence for this proposal using the [[0555 Gravitational path integral|gravitational path integral]] and the coding interpretation of holography, and we explain how similar effects arise in black hole physics in appropriate circumstances.\] # Hartman (Jul, a) ## Conformal Turaev-Viro Theory \[Links: [arXiv](https://arxiv.org/abs/2507.11652), [PDF](https://arxiv.org/pdf/2507.11652)\] \[Abstract: We define and study Conformal Turaev-Viro (CTV) theory, a dual formulation of [[0596 Virasoro TQFT|Virasoro TQFT]] based on triangulating 3-manifolds with tetrahedra. Edges of the triangulation are labeled by continuous conformal weights, and tetrahedra are glued together weighted by the Cardy density of states. We demonstrate that the CTV partition function is equal to the modular $S$-transform of the Virasoro TQFT amplitude-squared, $|Z_{Vir}|^2$. This is analogous to a known result for discrete spin networks. The derivation uses a variant of the chain-mail formalism, adapted to the Virasoro context. As a CFT application, we derive formulae for the $S$-transforms of the squared Virasoro crossing kernels. These results lay the topological foundation to study the exact path integral of pure AdS$_3$ quantum gravity by triangulations.\] # Hartman (Jul, b) ## Triangulating quantum gravity in AdS$_3$ \[Links: [arXiv](https://arxiv.org/abs/2507.12696), [PDF](https://arxiv.org/pdf/2507.12696)\] \[Abstract: The path integral of pure [[0002 3D gravity|3D gravity]] with negative cosmological constant is formulated on a finite region of spacetime $M$, with boundary conditions that fix geodesic lengths or dihedral angles on $\partial M$. In the dual CFT, this quasi-local amplitude calculates corrections to the Gaussian [[0663 OPE statistics|ensemble of OPE coefficients]] for black hole states. By triangulating $M$ with generalized tetrahedra, we develop a general method to construct semiclassical geometries and to calculate the exact gravitational path integral on a fixed hyperbolic topology. The path integral with fixed-length boundary conditions is a Virasoro TQFT amplitude-squared, and with fixed-angle boundary conditions it is a partition function of Conformal Turaev-Viro theory. The two are related by a modular $S$-transform. In addition, we show how to translate the calculation of OPE statistics from [[0596 Virasoro TQFT|Virasoro TQFT]] to the metric formalism, on general topologies. These results are derived exactly, and some examples are also checked semiclassically, including the geometries dual to the Virasoro $6j$-symbol and the modular $S$-matrix. The classical saddlepoint geometries are finite-volume hyperbolic 3-manifolds ending on pleated Riemann surfaces, which have vanishing extrinsic curvature except on geodesics where they can bend into corners. The hyperbolic volumes of these geometries match the predictions of Conformal Turaev-Viro theory and the dual CFT.\] # Harvey, Jensen, Uzu ## Comparing top-down and bottom-up holographic defects and boundaries \[Links: [arXiv](https://arxiv.org/abs/2504.13244), [PDF](https://arxiv.org/pdf/2504.13244)\] \[Abstract: In this work we consider domain walls and end-of-the-world branes in AdS/CFT, holographically dual to codimension-one conformal [[0065 Defect CFT|defects]] and conformal [[0548 Boundary CFT|boundaries]] respectively. In this setting there is an analogue of the ''[[0128 Bulk point singularity|bulk point]]'' singularity in boundary correlation functions, which we use to compare top-down and bottom-up constructions of these systems. For example, for a range of parameters the D3/D5 boundary CFT cannot be imitated by a tensionful end-of-the-world brane coupled to Einstein gravity, and in another range it can be modeled with a negative tension brane. Along the way we compute the central charge $b$ for the M2/M5 boundary CFT.\] # Haupfear, Martin, Svesko, Zukowski ## Untangling Selberg from the Wilson spool: 1-loop determinants and trace formulae in (A)dS$_{3}$ \[Links: [arXiv](https://arxiv.org/abs/2507.05358), [PDF](https://arxiv.org/pdf/2507.05358)\] \[Abstract: Leading quantum effects in perturbative quantum gravity are captured by functional determinants of kinetic operators. We study such 1-loop determinants in three-dimensional Euclidean (anti-) de Sitter gravity evaluated using two seemingly disparate tools, the Selberg zeta function and the [[0653 Wilson spool|Wilson spool]]. For the Euclidean BTZ black hole, we demonstrate the Wilson spool for massive bosons of arbitrary spin directly equates to a representation-theoretic version of the Selberg zeta function. In the case of Euclidean de Sitter, we show a new trace formula associated with the Fredholm determinant for the scalar Laplacian on the three-sphere reproduces the Wilson spool. Generalizing the trace formula, we comment on how to extend this Wilson spool construction to lens space quotients and higher-dimensional spheres.\] # Higginbotham ## On tests for baby universes in AdS/CFT \[Links: [arXiv](https://arxiv.org/abs/2507.05337), [PDF](https://arxiv.org/pdf/2507.05337)\] \[Abstract: To address a puzzle by Antonini and Rath -- where a single CFT state has two bulk duals, one with a [[0051 Baby universes|baby universe]] and one without -- Engelhardt and Gesteau recently devised a test for baby universes in AdS/CFT. Using the extrapolate dictionary, they showed that the boundary dual of a bulk swap test favored bulk spacetimes without a baby universe, providing evidence against their semiclassical validity. However, recent work suggests that holographic maps should post-select on such closed universes, and we argue that this is consistent with the extrapolate dictionary. We therefore construct a new holographic map for bulk states with baby universes and use this to show that the swap test cannot distinguish between Antonini and Rath's two candidate bulk duals. This not only allows for a valid semiclassical description of the baby universe, but also enables the application of recent techniques for including observers in holographic maps.\] # Hollands, Longo ## A New Proof of the QNEC \[Links: [arXiv](https://arxiv.org/abs/2503.04651), [PDF](https://arxiv.org/pdf/2503.04651)\] \[Abstract: We give a simplified proof of the [[0405 Quantum null energy condition|quantum null energy condition]] (QNEC). Our proof is based on an explicit formula for the shape derivative of the [[0199 Relative entropy|relative entropy]], with respect to an entangling cut. It allows bypassing the analytic continuation arguments of a previous proof by Ceyhan and Faulkner and can be used e.g., for defining entropy current fluctuations.\] # Hung, Jiang, Lao ## Universal Structures and Emergent Geometry from Large-$c$ BCFT Ensemble \[Links: [arXiv](https://arxiv.org/abs/2504.21660), [PDF](https://arxiv.org/pdf/2504.21660)\] \[Abstract: In this paper, we study the [[0154 Ensemble averaging|ensemble average]] of [[0548 Boundary CFT|boundary CFT]] (BCFT) data consistent with the [[0036 Conformal bootstrap|bootstrap]] equations. We apply the results to computing ensemble average of copies of multi-point correlation functions of boundary changing operators (BCO), and find the results in agreement with one copy of the [[0596 Virasoro TQFT|Virasoro TQFT]]. Further, we consider ensemble average of CFT path-integrals expressed as [[0054 Tensor network|tensor networks]] of BCO correlation functions using the formalism developed in [arXiv:2210.12127](https://arxiv.org/abs/2210.12127), [arXiv:2311.18005](https://arxiv.org/abs/2311.18005) and [arXiv:2403.03179](https://arxiv.org/abs/2403.03179). We find a natural emergence of locality and a loop-sum structure reminiscent of lattice integrable models. We illustrate this universal structure through explicit examples at genus zero and genus one. Moreover, we provide strong evidence that, at leading order in large-$c$, the results match those of [[0002 3D gravity|three-dimensional Einstein gravity]]. In the presence of closed CFT operator insertions, generalized free fields emerge, with their correlation functions governed by the shortest paths connecting the insertions.\] # Iizuka, Nishida ## Genuine multi-entropy and holography \[Links: [arXiv](https://arxiv.org/abs/2502.07995), [PDF](https://arxiv.org/pdf/2502.07995)\] \[Abstract: Is bipartite entanglement sufficient for holography? Through the analysis of the Markov gap, it is known that the answer is no. In this paper, we give a new perspective on this issue from a different angle using a multi-entropy. We define a genuine $\mathtt{q}$-partite multi-entropy from a $\mathtt{q}$-partite multi-entropy by subtracting appropriate linear combinations of $\mathtt{\tilde{q}}$-partite multi-entropies for $\mathtt{\tilde{q}} < \mathtt{q}$, in such a way that the genuine $\mathtt{q}$-partite multi-entropy vanishes for all $\mathtt{\tilde{q}}$-partite entangled states. After studying several aspects, we apply it to black holes and holography. For the application to black holes, we see that such a genuine $\mathtt{q}$-partite multi-entropy is important only after the Page time. For the application to holography, we prove that non-bipartite multi-entropies are always positive and $\mathcal{O}\left({1/ G_N}\right)$, as long as boundary subregions are connected. This indicates that for holography, genuine [[0264 Multi-partite entanglement|multi-partite entanglement]] is not small and plays an important role.\] # Kawamoto, Maeda, Nakamura, Takayanagi ## Traversable AdS Wormhole via Non-local Double Trace or Janus Deformation \[Links: [arXiv](https://arxiv.org/abs/2502.03531), [PDF](https://arxiv.org/pdf/2502.03531)\] \[Abstract: We study (i) Janus deformations and (ii) non-local double trace deformations of a pair of CFTs, as two different ways to construct CFT duals of [[0083 Traversable wormhole|traversable AdS wormholes]]. First, we construct a simple model of traversable wormholes by gluing two Poincaré AdS geometries and [[0086 Banados-Teitelboim-Zanelli black hole|BTZ]] black holes and compute holographic two point functions and ([[0052 Pseudo-entropy|pseudo]]) [[0301 Entanglement entropy|entanglement entropy]]. We point out that a Janus gravity solution describes a traversable wormhole when the deformation parameter takes imaginary values. On the other hand, we show that double trace deformations between two decoupled CFTs can reproduce two point functions of traversable AdS wormholes. By considering the case where the double trace deformation is given by a non-local [[0170 TTbar|TTbar]] deformation, we analyze the dual gravity which implies emergence of wormholes. We present toy model of these deformed CFTs by using free scalars and obtain qualitative behaviors expected for them. We argue that the crucial difference between the two constructions is that a global time slice of wormhole is described by a pure state for Janus deformations, while it is a mixed state for the double trace deformations.\] # Khan ## Conformal Cauchy Slice Holography: An Alternative Phase Space For Gravity \[Links: [arXiv](https://arxiv.org/abs/2507.14517), [PDF](https://arxiv.org/pdf/2507.14517)\] \[Abstract: The phase space of gravitational theories in asymptotically Anti-de Sitter (AAdS) spacetimes consists of geometries, matter configurations, and their conjugate momenta on a Cauchy surface, subject to the Hamiltonian, momentum, and matter-gauge constraints. When a unique maximal volume slice exists in all classical solutions of the bulk equations of motion, and the matter fields satisfy certain conditions, we show that this phase space is physically equivalent to an alternative phase space in which the Hamiltonian constraint is replaced by the real Weyl-anomaly constraint, while the momentum and matter-gauge constraints remain unchanged. A necessary requirement for a functional of the metric and matter configurations to qualify as a valid quantum gravity state is that it satisfies the operator gauge constraints. Partition functions of certain conformal field theories with imaginary central charge, defined on bulk Cauchy slices, satisfy these operator gauge constraints and therefore provide candidate quantum gravity states in the alternative phase space formulation.\] # Kibe, Roy ## Quantum null energy condition in quenched 2d CFTs \[Links: [arXiv](https://arxiv.org/abs/2503.17448), [PDF](https://arxiv.org/pdf/2503.17448)\] \[Abstract: The [[0405 Quantum null energy condition|quantum null energy condition]] (QNEC) is a lower bound on the expectation value of the null-null component of the energy-momentum tensor in terms of null variations of the entanglement entropy. A stronger version of the QNEC (the primary QNEC) is expected to hold in [[0003 2D CFT|1+1 dimensional conformal field theories]] (CFT). QNEC has been shown to impose non-trivial quantum thermodynamic restrictions on irreversible entropy production in quenches in 1+1 dimensional holographic CFTs. It is therefore natural to study if QNEC imposes similar bounds in other quench setups. In this paper we study QNEC in the Calabrese-Cardy global and local joining [[0558 Quantum quench|quenches]] using standard CFT techniques. In the global quench we show that the primary QNEC must hold at sufficiently early times and find that it imposes bounds on the four point correlators of twist fields in a boundary state. This is a constraint on the set of boundary states that satisfy the primary QNEC. Furthermore, we find that a violation of the primary QNEC implies a violation of the [[0417 Averaged null energy condition|averaged null energy condition]] (ANEC) in a conformally transformed frame. In the local quench we find similar bounds on four point correlators from both the primary and the usual QNEC.\] # Kumar ## Non-linear equation of motion for higher curvature semiclassical gravity \[Links: [arXiv](https://arxiv.org/abs/2501.10527), [PDF](https://arxiv.org/pdf/2501.10527)\] \[Abstract: We derive the non-linear semiclassical equation of motion for a general diffeomorphism-invariant theory of gravity by [[0302 Gravity from entanglement|leveraging the thermodynamic properties]] of closed causal horizons. Our work employs two complementary approaches. The first approach utilizes perturbative quantum gravity applied to a Rindler horizon. The result is then mapped to a stretched light cone, which can be understood as a union of Rindler planes. Here, we adopt the semiclassical physical process formulation, encapsulated by $\langle Q\rangle = T \delta S_{gen}$ where the heat-flux $\langle Q\rangle$ is related to the expectation value of stress-energy tensor $T_{ab}$ and $S_{gen}$ is the [[0004 Black hole entropy|generalized entropy]]. The second approach introduces a "higher curvature" [[0408 Raychaudhuri equation|Raychaudhuri equation]], where the vanishing of the quantum expansion $(\Theta)$ pointwise as required by restricted quantum focusing establishes an equilibrium condition, $(\delta S_{\text{gen}} = 0)$, at the null boundary of a causal diamond. While previous studies have only derived the linearized semiclassical equation of motion for [[0006 Higher-derivative gravity|higher-curvature gravity]], our work resolves this limitation by providing a fully non-linear formulation without invoking holography.\] # Kusuki, Ooguri, Pal ## Universality of Rényi Entropy in Conformal Field Theory \[Links: [arXiv](https://arxiv.org/abs/2503.24353), [PDF](https://arxiv.org/pdf/2503.24353)\] \[Abstract: We use the thermal effective theory to prove that, for the vacuum state in any conformal field theory in $d$ dimensions, the $n$-th [[0293 Renyi entropy|Rényi entropy]] $S_A^{(n)}$ behaves as $S_A^{(n)} = \frac{f}{(2\pi n)^{d-1}} \frac{ {\rm Area}(\partial A)}{(d-2)\epsilon^{d-2}}\left(1+O(n)\right)$ in the $n \rightarrow 0$ limit when the boundary of the entanglement domain $A$ is spherical with the UV cutoff $\epsilon$. The theory dependence is encapsulated in the cosmological constant $f$ in the thermal effective action. Using this result, we estimate the density of states for large eigenvalues of the [[0416 Modular Hamiltonian|modular Hamiltonian]] for the domain $A$. In two dimensions, we can use the hot spot idea to derive more powerful formulas valid for arbitrary positive $n$. We discuss the difference between two and higher dimensions and clarify the applicability of the hot spot idea. We also use the thermal effective theory to derive an analog of the [[0406 Cardy formula|Cardy formula]] for boundary operators in higher dimensions.\] # Maloney, Meruliya, Van Raamsdonk ## Ordinary wormholes \[Links: [arXiv](https://arxiv.org/abs/2503.12227), [PDF](https://arxiv.org/pdf/2503.12227)\] \[Abstract: [[0278 Euclidean wormholes|Euclidean wormholes]] have played a key role in the recent ''disorder averaged" approaches to quantum gravity and holography, but are typically only considered in somewhat special theories of gravity, such as theories in low dimensions or theories with exotic matter content (such as axions). These exotic theories have advantage that both the matter and gravitational sectors can be treated completely classically. However, once this constraint is relaxed we find that Euclidean wormholes arise generically, with no special constraints on the matter content. The key point is that there is a self-consistent approximation where the metric is treated classically but matter is treated quantum mechanically. The resulting wormholes are *ordinary* in the sense that they rely on the usual approximations used in, for example, the construction of star or FRW solutions in general relativity. Indeed, these are the Euclidean continuations of ordinary FRW solutions with big bang/crunch singularities. We describe several examples of these ordinary wormholes and discuss the relation to existing constructions and the holographic interpretation in terms of a dual CFT.\] # Mei, Mo ## Soft Photon, Gluon and Graviton Theorems in (A)dS from Conformal Invariance \[Links: [arXiv](https://arxiv.org/abs/2506.11766), [PDF](https://arxiv.org/pdf/2506.11766)\] \[Abstract: We present new [[0009 Soft theorems|soft theorems]] for photon, gluon, and graviton correlators at tree level in (Anti)-de Sitter space. The results are derived by applying conformal [[0106 Ward identity|Ward identities]] to constrain the structure of [[0109 Witten diagrams|Witten diagrams]].\] # Milekhin, Adamska, Preskill ## Observable and computable entanglement in time \[Links: [arXiv](https://arxiv.org/abs/2502.12240), [PDF](https://arxiv.org/pdf/2502.12240)\] \[Abstract: We propose a novel family of entanglement measures for [[0606 Timelike entanglement|time-separated]] subsystems. Our definitions are applicable to any quantum system, continuous or discrete. To illustrate their utility, we derive upper and lower bounds on time-separated correlation functions, akin to the bound on spatially separated correlators in terms of the [[0300 Mutual information|mutual information]]. In certain cases our bounds are tight. For relativistic quantum field theories our definition agrees with the analytic continuation from spacelike to timelike separated regions. We provide relevant measurement protocols and execute them on the IBM quantum device $\texttt{ibm\_sherbrooke}$ for a simple qubit system. Also we perform explicit computations for an Ising spin chain, free fermions, (1+1)-dimensional [[0003 2D CFT|conformal field theories]] and [[0001 AdS-CFT|holographic]] theories. Finally we explain how the proposed entanglement in time provides a microscopic definition for the recently introduced timelike [[0052 Pseudo-entropy|pseudoentropy]].\] # Miyachi, Namba, Omiya, Oshita ## Path to an exact WKB analysis of black hole quasinormal modes \[Links: [arXiv](https://arxiv.org/abs/2503.17245), [PDF](https://arxiv.org/pdf/2503.17245)\] \[Abstract: We investigate black hole [[0325 Quasi-normal modes|quasinormal modes]] using the exact WKB method. We perform an analytic continuation from the horizon to infinity along the positive real axis of the radial coordinate and impose appropriate boundary conditions at these asymptotic positions. We clarify the role of previously overlooked logarithmic spirals of Stokes curves and branch cuts emerging from the horizon. We carefully reformulate the derivation of the quasinormal mode conditions using the exact WKB analysis, incorporating the contributions from these features into the calculation. We successfully derive correct results for both solvable model examples and the Schwarzschild spacetime. Our formulation enjoys straightforward extensions to other background geometries as well as a wide range of other physical systems.\] # Miyaji, Mori, Okuyama ## Finite $N$ Bulk Hilbert Space in ETH Matrix Model for double-scaled SYK \[Links: [arXiv](https://arxiv.org/abs/2505.13194), [PDF](https://arxiv.org/pdf/2505.13194)\] \[Abstract: We extend the notion of chord number in the strict large $N$ [[0503 Double-scaled SYK|double-scaled Sachdev-Ye-Kitaev]] (DSSYK) model to the corresponding finite $N$ ETH [[0197 Matrix model|matrix model]]. The chord number in the strict large $N$ DSSYK model is known to correspond to the discrete length of the Einstein-Rosen bridge in the gravity dual, which reduces to the renormalized geodesic length in [[0050 JT gravity|JT gravity]] at weak coupling. At finite $N$, these chord number states can be constructed but form an overcomplete basis of the non-perturbative Hilbert space, as the structure of the inner product gets significantly modified due to the Cayley-Hamilton theorem: There are infinitely many null states. In this paper, by considering ''EFT for gravitational observables'' or a version of ''[[0576 Non-isometric codes|non-isometric code]]'', we construct a family of chord number operators at finite $N$. While the constructed chord number operator depends on the reference chord number state, it realizes approximate $q$-deformed oscillator algebra and reproduces semiclassical bulk geometry around the reference state. As a special case, we will show that when the reference is chosen to be the chord number zero state, the chord number operator precisely matches with the [[0564 Krylov complexity|Krylov state complexity]], leading to the ''ramp-slope-plateau'' behavior at late times, implying the formation of ''gray hole''.\] # Miyaji, Ruan, Shibuya, Yano ## Non-perturbative Overlaps in JT Gravity: From Spectral Form Factor to Generating Functions of Complexity \[Links: [arXiv](https://arxiv.org/abs/2502.12266), [PDF](https://arxiv.org/pdf/2502.12266)\] \[Abstract: The interplay between black hole interior dynamics and [[0008 Quantum chaos|quantum chaos]] provides a crucial framework for probing quantum effects in quantum gravity. In this work, we investigate non-perturbative overlaps in [[0050 JT gravity|Jackiw-Teitelboim (JT) gravity]] to uncover universal signatures of quantum chaos and [[0204 Quantum complexity|quantum complexity]]. Taking advantage of universal spectral correlators from [[0579 Random matrix theory|random matrix theory]], we compute the overlaps between the [[0574 Thermofield double|thermofield double]] (TFD) state and two distinct classes of states: fixed-length states, which encode maximal volume slices, and time-shifted TFD states. The squared overlaps naturally define probability distributions that quantify the expectation values of gravitational observables. Central to our results is the introduction of generating functions for quantum complexity measures, such as $\langle e^{-\alpha \ell} \rangle$. The time evolution of these generating functions exhibits the universal slope-ramp-plateau structure, mirroring the behavior of the [[0062 Spectral form factor|spectral form factor]] (SFF). Using generating functions, we further demonstrate that the universal time evolution of complexity for chaotic systems, which is characterized by a linear growth followed by a late-time plateau, arises from the disappearance of the linear ramp as the regularization parameter $\alpha$ decreases. With regard to the time-shifted TFD state, we derive a surprising result: the expectation value of the time shift, which classically grows linearly, vanishes when non-perturbative quantum corrections are incorporated. This cancellation highlights a fundamental distinction between semiclassical and quantum gravitational descriptions of the black hole interior. All our findings establish generating functions as powerful probes of quantum complexity and chaos in gravitational and quantum systems.\] # Narovlansky ## Towards a microscopic description of de Sitter dynamics \[Links: [arXiv](https://arxiv.org/abs/2506.02109), [PDF](https://arxiv.org/pdf/2506.02109)\] \[Abstract: Describing dynamics in a gravitational universe with positive cosmological constant, such as [[0545 de Sitter quantum gravity|de Sitter space]], is a conceptually challenging problem. We propose a principle for constructing a quantum system that can potentially be used to study this question. This quantum system describes a heavy object in such a universe interacting with its environment, to which gauge invariant dynamical observables can be anchored. In order to describe gravity with positive cosmological constant, the proposed quantum system needs to agree with all known semiclassical results. We investigate this with a particular microscopic realization constructed using [[0201 Sachdev-Ye-Kitaev model|SYK]]. We first find that correlators match the classical limit of gravity, given by quantum fields in rigid de Sitter space. In particular, the usual UV behavior of quantum fields is surprisingly reproduced by the quantum mechanical system. In order to probe small effects in the gravitational constant, we also consider the intrinsically dynamical [[0482 Out-of-time-order correlator|out-of-time-order correlators]] (OTOCs). These correspond to gravitational scattering in the bulk away from the worldline associated with the quantum system. Such OTOCs have highly non-trivial features in de Sitter space, including a [[0466 Lyapunov exponent|Lyapunov exponent]] which is twice as big as the maximal chaos exponent from the [[0474 Chaos bound|bound on chaos]], as well as an unusual behavior of the coefficients in various OTOCs. Interestingly, we find that these features are reproduced by the quantum system.\] # Nomura, Ugajin ## Nonperturbative Quantum Gravity in a Closed Lorentzian Universe \[Links: [arXiv](https://arxiv.org/abs/2505.20390), [PDF](https://arxiv.org/pdf/2505.20390)\] \[Abstract: We study how meaningful physical predictions can arise in nonperturbative quantum gravity in a [[0632 Closed universe|closed Lorentzian universe]]. In such settings, recent developments suggest that the quantum gravitational Hilbert space is one-dimensional and real for each $\alpha$-sector, as induced by spacetime wormholes. This appears to obstruct the conventional quantum-mechanical prescription of assigning probabilities via projection onto a basis of states. While previous approaches have introduced external observers or augmented the theory to resolve this issue, we argue that quantum gravity itself contains all the necessary ingredients to make physical predictions. We demonstrate that the emergence of classical observables and probabilistic outcomes can be understood as a consequence of partial observability: physical observers access only a subsystem of the universe. Tracing out the inaccessible degrees of freedom yields reduced density matrices that encode classical information, with uncertainties exponentially suppressed by the environment's entropy. We develop this perspective using both the Lorentzian path integral and operator formalisms and support it with a simple microscopic model. Our results show that quantum gravity in a closed universe naturally gives rise to meaningful, robust predictions without recourse to external constructs.\] # Rabinovici, Sanchez-Garrido, Shir, Sonner ## Krylov Complexity \[Links: [arXiv](https://arxiv.org/abs/2507.06286), [PDF](https://arxiv.org/pdf/2507.06286)\] \[Abstract: We introduce and review a new complexity measure, called '[[0564 Krylov complexity|Krylov complexity]]', which takes its origins in the field of [[0008 Quantum chaos|quantum-chaotic]] dynamics, serving as a canonical measure of operator growth and spreading. Krylov complexity, underpinned by the Lanczos algorithm, has since evolved into a highly diverse field of its own right, both because of its attractive features as a complexity, whose definition does not depend on arbitrary control parameters, and whose phenomenology serves as a rich and sensitive probe of chaotic dynamics up to exponentially late times, but also because of its relevance to seemingly far-afield subjects such as holographic dualities and the quantum physics of black holes. In this review we give a unified perspective on these topics, emphasizing the robust and most general features of K-complexity, both in chaotic and integrable systems, state and prove theorems on its generic features and describe how it is geometrised in the context of (dual) gravitational dynamics. We hope that this review will serve both as a source of intuition about K-complexity in and of itself, as well as a resource for researchers trying to gain an overview over what is by now a rather large and multi-faceted literature. We also mention and discuss a number of open problems related to K-complexity, underlining its currently very active status as a field of research.\] # Roy, Lukyanov, Saleur ## Boundary Conditions for the Entanglement Cut in 2D Conformal Field Theories \[Links: [arXiv](https://arxiv.org/abs/2503.12674), [PDF](https://arxiv.org/pdf/2503.12674)\] \[Abstract: The entanglement spectra for a subsystem in a spin chain fine-tuned to a quantum-critical point contains signatures of the underlying quantum field theory that governs its low-energy properties. For an open chain with given boundary conditions described by a [[0003 2D CFT|2D conformal field theory]]~(CFT), the entanglement spectrum of the left/right half of the system coincides with a [[0548 Boundary CFT|boundary CFT]] spectrum, where one of the boundary conditions arise due to the 'entanglement cut'. The latter has been argued to be conformal and has been numerically found to be the 'free' boundary condition for Ising, Potts and free boson theories. For these models, the 'free' boundary condition for the lattice degree of freedom has a counterpart in the continuum theory. However, this is not true in general. Here, this question is analyzed for the unitary [[0599 Minimal models|minimal models]] of 2D CFTs using the density matrix renormalization group technique. The entanglement spectra are computed for blocks of spins in open chains of A-type restricted solid-on-solid models with identical boundary conditions at the ends. The imposed boundary conditions are realized exactly for these lattice models due to their integrable nature. The obtained entanglement spectra are in good agreement with certain boundary CFT spectra. The boundary condition for the entanglement cut is found to be conformal and to coincide with the one with the highest boundary entropy. This identification enables determination of the exponents governing the unusual corrections to the [[0301 Entanglement entropy|entanglement entropy]] from the CFT partition functions. These are compared with numerical results.\] # Shi, Turiaci ## The phase of the gravitational path integral \[Links: [arXiv](https://arxiv.org/abs/2504.00900), [PDF](https://arxiv.org/pdf/2504.00900)\] \[Abstract: The [[0555 Gravitational path integral|gravitational path integral]] on $S^2 \times S^2$ can be interpreted either as evaluating a contribution to the norm of the [[0162 No-boundary wavefunction|Hartle-Hawking wavefunction]] conditional on spatial $S^1 \times S^2$ topology, or the pair creation rate of black holes in de Sitter. Both interpretations are distinguished at the quantum level. The former requires the path integral to be real and the latter to be imaginary. We develop a formalism to efficiently compute the phase of the gravitational path integral on Einstein spaces. We apply it to a broad class of spacetimes and in particular $S^2\times S^{D-2}$, finding it to be real and positive. We generalize some of the analysis to cases with charge and rotation.\] # Turiaci, Wu ## The wavefunction of a quantum $S^1 \times S^2$ universe \[Links: [arXiv](https://arxiv.org/abs/2503.14639), [PDF](https://arxiv.org/pdf/2503.14639)\] \[Abstract: We study quantum gravity corrections to the [[0162 No-boundary wavefunction|no-boundary wavefunction]] describing a universe with spatial topology $S^1\times S^2$. It has been suggested that quantum effects become increasingly important when the size of the circle is large relative to the sphere. In this paper, we confirm this claim by an explicit four-dimensional one-loop calculation of the gravitational path integral preparing such a state. In the process, we clarify some aspects of the [[0555 Gravitational path integral|gravitational path integral]] on [[0335 Complex metrics|complex spacetimes]]. These quantum corrections play a crucial role in ensuring that the norm of the wavefunction is naturally expressed in terms of a path integral over $S^2 \times S^2$ at the classical level. We extend some of the analysis to more general spatial topologies, as well as to the inclusion of fermions.\] # Wang ## Microscopic origin of the entropy of de Sitter spacetime \[Links: [arXiv](https://arxiv.org/abs/2506.03058), [PDF](https://arxiv.org/pdf/2506.03058)\] \[Abstract: We construct an infinite family of semiclassical de Sitter (dS) microstates, realized as backreacted geometries of dS spacetime with a constant tension thin-shell brane located outside the dS event horizon. The contribution of wormholes in semiclassical [[0555 Gravitational path integral|gravitational path integral]] computations of overlaps between these [[0248 Black hole microstates|microstates]] implies that they are not fully orthogonal. We count the dimension of the Hilbert space spanned by these dS microstates and find that it equals the exponential of the Gibbons-Hawking entropy of the de Sitter spacetime. Our construction provides a gravitational states counting derivation of Gibbons-Hawking formula for dS entropy.\] # Wang, Wang, Wei ## Wormholes with Ends of the World \[Links: [arXiv](https://arxiv.org/abs/2504.12278), [PDF](https://arxiv.org/pdf/2504.12278)\] \[Abstract: We study classical wormhole solutions in [[0002 3D gravity|3D gravity]] with end-of-the-world (EOW) branes, conical defects, kinks, and punctures. These solutions compute statistical averages of an [[0154 Ensemble averaging|ensemble]] of [[0548 Boundary CFT|boundary conformal field theories]] (BCFTs) related to universal asymptotics of OPE data extracted from 2D [[0036 Conformal bootstrap|conformal bootstrap]]. Conical defects connect BCFT bulk operators; branes join BCFT boundary intervals with identical boundary conditions; kinks (1D defects along branes) link BCFT boundary operators; and punctures (0D defects) are endpoints where conical defects terminate on branes. We provide evidence for a correspondence between the gravity theory and the ensemble. In particular, the agreement of $g$-function dependence results from an underlying topological aspect of the on-shell EOW brane action, from which a BCFT analogue of the Schlenker-Witten theorem also follows.\] # Yan ## Puzzles in 3D Off-Shell Geometries via VTQFT \[Links: [arXiv](https://arxiv.org/abs/2502.16686), [PDF](https://arxiv.org/pdf/2502.16686)\] \[Abstract: We point out a difficulty with a naive application of [[0596 Virasoro TQFT|Virasoro TQFT]] methods to compute path integrals for two types of off-shell 3-dimensional geometries. Maxfield-Turiaci proposed solving the negativity problem of pure [[0002 3D gravity|3d gravity]] by summing over off-shell geometries known as Seifert manifolds. We attempt to compute Seifert manifolds using Virasoro TQFT. Our results don't match completely with Maxfield-Turiaci. We trace the discrepancies to not including the mapping class group properly. We also compute a 3-boundary torus-wormhole by extrapolating from an on-shell geometry. We encounter challenges similar to those observed in the comparison between the genuine off-shell computation of a torus-wormhole by Cotler-Jensen and the extrapolation from an on-shell configuration.\] # Yang, Zhang, Zheng ## Comments on the de Sitter Double Cone \[Links: [arXiv](https://arxiv.org/abs/2505.08647), [PDF](https://arxiv.org/pdf/2505.08647)\] \[Abstract: We study the double cone geometry proposed by Saad, Shenker, and Stanford in de Sitter space. We demonstrate that with the inclusion of static patch observers, the double cone leads to a linear ramp consistent with random matrix behavior. This ramp arises from the relative time shift between two clocks located in opposite static patches.\] # Zolfi, Alishahiha ## On JT gravity path integrals and the tunneling process \[Links: [arXiv](https://arxiv.org/abs/2502.14448), [PDF](https://arxiv.org/pdf/2502.14448)\] \[Abstract: The [[0050 JT gravity|Jackiw-Teitelboim (JT) gravity]] path integral of the trumpet can be interpreted as a transition amplitude from an older black hole to a younger one, accompanied by the emission of a [[0051 Baby universes|baby universe]], represented by the geodesic boundary of the trumpet. However, this interpretation becomes less straightforward for geometries with higher genus and multiple geodesic boundaries. In this paper, we examine the path integral for these more complex geometries and find that maintaining this interpretation requires accounting for a portion of the moduli space.\]