Modular Berry transport is a useful way to understand how geometric bulk information is encoded in the boundary CFT: the modular curvature is directly related to the bulk Riemann curvature. We extend this approach by studying modular transport in the presence of a non-trivial quantum extremal surface. Focusing on JT gravity on an AdS background coupled to a non-gravitating bath, we compute the modular curvature of an interval in the bath in the presence of an island: the Quantum Extremal Modular Curvature (QEMC). We highlight some important properties of the QEMC, most importantly that it is non-local in general. In an OPE limit, the QEMC becomes local and probes the bulk Riemann curvature in regions with an island. Our work gives a new approach to probe physics behind horizons.

United States

Lars Aalsma

Cynthia Keeler

Claire Zukowski

Quantum extremal modular curvature: modular transport with islands

A correction to this paper has been published: https://doi.org/10.1007/JHEP07(2023)120

United Kingdom

Alejandra Castro

Netherlands

Japan

Ioana Coman

Jackson Fliss

Erratum to: Keeping matter in the loop in dS3 quantum gravity

We propose a mechanism that couples matter fields to three-dimensional quantum gravity, which can be used for theories with a positive or negative cosmological constant. Our proposal is rooted in the Chern-Simons formulation of three-dimensional gravity and makes use of the Wilson spool, a collection of Wilson loops winding around closed paths of the background. We show that the Wilson spool correctly reproduces the one-loop determinant of a free massive scalar field on rotating black holes in <math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math> and Euclidean <math><mrow><msub><mrow><mi>dS</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math> as <math><msub><mi>G</mi><mi>N</mi></msub><mo>→</mo><mn>0</mn></math>. Moreover, we describe how to incorporate quantum metric fluctuations into this formalism.

Jackson R. Fliss

Coupling Fields to 3D Quantum Gravity via Chern-Simons Theory

We present a new perspective on bulk reconstruction using Berry phases in the boundary conformal field theory (CFT). Our parallel transport of modular Hamiltonians is associated to a trajectory in the space of states, which we obtain from the insertion of a source in the Euclidean path integral. Using a modular version of the extrapolate dictionary and the equivalence between modular flow in the boundary and the bulk, we show that the expectation value of the modular Berry curvature on the boundary agrees with an appropriately defined bulk symplectic form associated to the entanglement wedge. In addition, we derive a quantum information metric on the space of density matrices from the Berry curvature, which is related to the canonical energy in the bulk. We also explore the case where a state change reduces to a shape change, uncovering the coadjoint orbit structure of kinematic space in higher dimensions.

China

Bartłomiej Czech

Jan de Boer

Ricardo Espíndola

Bahman Najian

Jeremy van der Heijden

Changing states in holography: From modular Berry curvature to the bulk symplectic form

We propose a mechanism that couples matter fields to three-dimensional de Sitter quantum gravity. Our construction is based on the Chern-Simons formulation of three-dimensional Euclidean gravity, and it centers on a collection of Wilson loops winding around Euclidean de Sitter space. We coin this object a Wilson spool. To construct the spool, we build novel representations of 𝔰𝔲(2). To evaluate the spool, we adapt and exploit several known exact results in Chern-Simons theory. Our proposal correctly reproduces the one-loop determinant of a free massive scalar field on S 3 as G  N  → 0. Moreover, allowing for quantum metric fluctuations, it can be systematically evaluated to any order in perturbation theory.

Keeping matter in the loop in dS3 quantum gravity

We study the parallel transport of modular Hamiltonians encoding entanglement properties of a state. In the case of 2d CFT, we consider a change of state through action with a suitable diffeomorphism on the circle: one that diagonalizes the adjoint action of the modular Hamiltonian. These vector fields exhibit kinks at the interval boundary, thus together with their central extension they differ from usual elements of the Virasoro algebra. The Berry curvature associated to state-changing parallel transport is the Kirillov-Kostant symplectic form on an associated coadjoint orbit, one which differs appreciably from known Virasoro orbits. We find that the boundary parallel transport process computes a bulk symplectic form for a Euclidean geometry obtained from the backreaction of a cosmic brane, with Dirichlet boundary conditions at the location of the brane. We propose that this gives a reasonable definition for the symplectic form on an entanglement wedge.

Jan Boer

Poland

Dimitrios Patramanis

Jeremy Heijden

Virasoro entanglement Berry phases

We study circuit complexity for conformal field theory states in an arbitrary number of dimensions. Our circuits start from a primary state and move along a unitary representation of the Lorentzian conformal group. Different choices of distance functions can be understood in terms of the geometry of coadjoint orbits of the conformal group. We explicitly relate our circuits to timelike geodesics in anti–de Sitter space and the complexity metric to distances between these geodesics. We extend our method to circuits in other symmetry groups using a group theoretic generalization of the notion of coherent states.

Nicolas Chagnet

Israel

Shira Chapman

Complexity for Conformal Field Theories in General Dimensions

We construct local probes in the static patch of Euclidean dS3 gravity. These probes are Wilson line operators, designed by exploiting the Chern-Simons formulation of 3D gravity. Our prescription uses non-unitary representations of so (4) ≃ su (2) L  × su(2) R , and we evaluate the Wilson line for states satisfying a singlet condition. We discuss how to reproduce the Green’s functions of massive scalar fields in dS3, the construction of bulk fields, and the quasinormal mode spectrum. We also discuss the interpretation of our construction in Lorentzian signature in the inflationary patch, via SL(2, ℂ) Chern-Simons theory.

Philippe Sabella-Garnier

Gravitational Wilson lines in 3D de Sitter

Kinematic space has been defined as the space of codimension-2 spacelike extremal surfaces in anti de Sitter (AdS d+1) spacetime which, by the Ryu-Takayanagi proposal, compute the entanglement entropy of spheres in the boundary CFT d . It has recently found many applications in holography. Coadjoint orbits are symplectic manifolds that are the classical analogues of a Lie group’s unitary irreducible representations. We prove that kinematic space is a particular coadjoint orbit of the d-dimensional conformal group SO(d, 2). In addition, we show that the Crofton form on kinematic space associated to AdS3, that was shown to compute the lengths of bulk curves, is equal to the standard Kirillov-Kostant symplectic form on the coadjoint orbit. Since kinematic space is Kähler in addition to symplectic, it can be quantized. The orbit method extends the kinematic space dictionary, which was originally motivated through connections to integral geometry, by directly translating geometrical properties of holographic auxiliary spaces into statements about the representation theory of the conformal group.

Robert Penna

Kinematic space and the orbit method

Recently, two groups have made distinct proposals for a de Sitter space that is emergent from conformal field theory (CFT). The first proposal is that, for two-dimensional holographic CFTs, the kinematic space of geodesics on a space-like slice of the asymptotically anti-de Sitter bulk is two-dimensional de Sitter space (dS 2 ), with a metric that can be derived from the entanglement entropy of intervals in the CFT. In the second proposal, de Sitter dynamics emerges naturally from the first law of entanglement entropy for perturbations around the vacuum state of CFTs. We provide support for the equivalence of these two emergent spacetimes in the vacuum case and beyond. In particular, we study the kinematic spaces of nontrivial solutions of 3d gravity, including the BTZ black string, BTZ black hole, and conical singularities. We argue that the resulting spaces are generically globally hyperbolic spacetimes that support dynamics given boundary conditions at future infinity. For the BTZ black string, corresponding to a thermal state of the CFT, we show that both prescriptions lead to an emergent hyperbolic patch of dS 2 . We offer a general method for relating kinematic space and the auxiliary de Sitter space that is valid in the vacuum and thermal cases.