We study geometric modular flows in two-dimensional conformal field theories. We explore which states exhibit a geometric 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 entanglement entropy. We also consider geometric flows beyond the Rindler wedge setting, and in higher dimensions.

Canada

Jacqueline Caminiti

Italy

Federico Capeccia

Luca Ciambelli

Robert Myers

Geometric modular flows in 2d CFT and beyond

We discuss the scattering of massive scalar probes off Myers-Perry black holes in the Kerr-Schild gauge. Extending the analysis performed recently for Kerr(-Newman) black holes, we show that the Kerr-Schild gauge allows to write down the tree-level scattering amplitude for Myers-Perry black holes in analytic form. For generic values of the angular momenta, Myers-Perry solutions have a richer multipolar structure compared to their four-dimensional counterparts, because they are characterized by the presence of stress multipoles, together with the more familiar mass and current multipoles. By focusing on the five-dimensional case, we derive the leading eikonal phase from the scattering amplitude and we give an explicit expression for two limiting scenari, namely when the two angular momenta are the same, so that the mass multipoles vanish but still the solution has a non-vanishing stress quadrupole and a current dipole, and when one of the two angular momenta is zero, and correspondingly the stress multipoles vanish similar to the Kerr case.

Massimo Bianchi

Claudio Gambino

Fabio Riccioni

Vincenzo Zevola

Probing the multipolar structure of Myers-Perry black holes with scattering amplitudes

Batia Friedman-Shaw

Alex May

France

Olga Papadoulaki

Erratum: Holographic scattering and non-minimal RT surfaces

We investigate the quasinormal modes of massive scalar fields in the background of five-dimensional Myers-Perry black holes. In particular, we explore the case for Myers-Perry black holes with two arbitrary rotation parameters. Since the Klein-Gordon equation for the scalar field is separable, we numerically compute the scalar quasinormal modes by using the radial and angular equations. Two methods, the continued fraction method and matrix method, are used in the numerical calculation. We find that all obtained modes have negative imaginary parts and are decaying modes. We also consider the impact of the rotation parameters, scalar field mass <italic>&#956;</italic> and azimuthal numbers on the scalar quasinormal modes. Besides, when the scalar mass <italic>&#956;</italic> becomes relatively large, we also find the long-living scalar modes. Our numerical results also demonstrate the symmetries of the QNMs explicitly.

China

Zi-Yang Huang

Jia-Hui Huang

Quasinormal modes of massive scalar fields in five-dimensional Myers-Perry black holes with two arbitrary rotation parameters

We revisit the calculation of vacuum entanglement entropy in free Maxwell theory in four-dimensional Minkowski spacetime. Weyl invariance allows for this theory to be embedded as a patch inside the Einstein static universe. We use conformal inversions to extend the conformal primary solutions of the equations of motion labelled by a boost-weight ∆ = 1 + $\textit{iλ}$ to an inverted Minkowski patch centered at spacelike infinity of the original patch. For $\textit{λ}$ ≠ 0 these solutions admit an expansion in terms of wavefunctions supported in the (future) Milne wedges of the original and inverted Minkowski patches, that diagonalize the respective Milne times. The Minkowski vacuum can then be described as a thermofield double state on these two Milne wedges. We characterize the soft sectors of $\textit{λ}$ = 0 modes supported in the two Milne wedges. Upon reinterpreting the non-pure gauge $\textit{λ}$ = 0 wavefunctions as sourced by image charges in the inverse Minkowski patch, we construct an appropriate entangling constraint in the Minkowski theory, thereby characterizing the physical state space. We show that the edge mode contribution to the vacuum entanglement entropy is due to correlations between the soft charges of the two Milne patches, or equivalently non-trivial conformally soft mode configurations at the entangling surface.

United States

Hong Chen

United Kingdom

Ana-Maria Raclariu

Entanglement, soft modes, and celestial CFT

What does a black hole look like? In 1 + 3 spacetime dimensions, the optical appearance of a black hole is a bidimensional region in the observer’s sky often called the black hole shadow, as supported by the EHT observations. In higher dimensions this question is more subtle and observational setup dependent. Previous studies considered the shadows of higher dimensional black holes to remain bidimensional. We argue that the latter should be regarded as a tomography of a higher dimensional structure, the $\textit{hypershadow}$, which would be the structure “seen” by higher dimensional observers. As a case study we consider the cohomogeneity-one Myers-Perry black hole in 1 + 4 dimensions, and compute its tridimensional hypershadow.

Portugal

João Novo

Pedro Cunha

Carlos Herdeiro

Hypershadows of higher dimensional black objects: a case study of cohomogeneity-one d=5 Myers-Perry

Hawking’s black hole area theorem provides a geometric realization of the second law of thermodynamics and constrains gravitational processes. In this work we explore a one-parameter extension of this constraint formulated in terms of the monotonicity properties of Rényi entropies. We focus on black hole mergers in asymptotically AdS space and determine new restrictions which these Rényi second laws impose on the final state. We evaluate the entropic inequalities starting from the thermodynamic ensembles description of black hole geometries, and find that for many situations they set more stringent bounds than those imposed by the area increase theorem.

Alice Bernamonti

Federico Galli

Netherlands

Ignacio Reyes

Rényi second laws for black holes

In the AdS/CFT correspondence, the causal structure of the bulk AdS spacetime is tied to entanglement in the dual CFT. This relationship is captured by the connected wedge theorem [1], which states that a bulk scattering process implies the existence of $\textit{O}$(1/$\textit{G}$ ) entanglement between associated boundary subregions. In this paper, we study the connected wedge theorem in two asymptotically AdS spacetimes: the conical defect and BTZ black hole geometries. In these settings, we find that bulk scattering processes require not just large entanglement, but also additional restrictions related to candidate RT surfaces which are non-minimal. We argue these extra relationships imply a certain CFT entanglement structure involving internal degrees of freedom. Because bulk scattering relies on sub-AdS scale physics, this supports the idea that sub-AdS scale locality emerges from internal degrees of freedom. While the new restriction that we identify on non-minimal surfaces is stronger than the initial statement of the connected wedge theorem, we find that it is necessary but still not sufficient to imply bulk scattering in mixed states.

Holographic scattering and non-minimal RT surfaces

We employ double holography to examine a system of two entangled gravitating universes that live on two codimension-one branes in an asymptotically AdS3 spacetime with two disjoint conformal boundaries. There are distinct brane configurations depending on the temperature of the thermofield double (TFD) state between the left and right systems. The topology transition between two branes is naturally identified with the emergence of an Einstein-Rosen bridge connecting the two entangled universes. This doubly holographic construction offers a holographic perspective on gravitational collapse and black hole formation in brane universes. Through this holographic framework, we analyze the quantum information structure of the two gravitating universes. Specifically, we calculate the mutual information between defects present in the boundary theories on the left and right sides. Furthermore, we investigate the decoupling process in the Hayden-Preskill protocol applied to the two copies of the defect field theory and discuss the interpretation of the Yoshida-Kitaev decoding protocol.

Japan

Shan-Ming Ruan

Tomonori Ugajin

Double holography of entangled universes

We evaluate the vacuum entanglement entropy across a cut of future null infinity for free Maxwell theory in four-dimensional Minkowski spacetime. The Weyl invariance of 4D Maxwell theory allows us to embed the Minkowski spacetime inside the Einstein static universe. The Minkowski vacuum can then be described as a thermofield double state on the (future) Milne wedges of the original and inverted Minkowski patches. We show that the soft mode contribution to entanglement entropy is due to correlations between asymptotic charges of these Milne wedges or, equivalently, nontrivial conformally soft (or edge) mode configurations at the entangling surface.

Hong Zhe Chen

Entanglement, soft modes, and celestial holography