Conformal Boundary Research Articles

Applied holography of the AdS5–Kerr space–time

Asymptotically anti-de Sitter–Kerr black holes (we focus here on the five-dimensional case) are associated holographically with matter at conformal infinity which has a nonzero angular momentum density. It is natural to attempt to associate this angular momentum with the recently discovered vorticity of the plasmas produced in peripheral heavy-ion collisions. We assume that an AdS5–Kerr black hole with angular momentum to mass ratio [Formula: see text] is dual to boundary matter with an angular momentum density to energy density ratio also equal to [Formula: see text]. With this assumption, we find that, for collisions corresponding to a given value of [Formula: see text], there is a maximal possible angular velocity (well below the maximal value permitted by causality) for such matter at infinity, and that this value is in approximate agreement with the experimentally reported value of the average plasma vorticity produced in typical peripheral collisions of heavy ions.

Hausdorff closed limits and the c-boundary II: null infinity and black holes

We have shown in a companion paper (e Silva et al (2019)) that when a spacetime is globally hyperbolic with a (possibly empty) smooth timelike boundary , then a metrizable topology, the closed limit topology (CLT), can be advantageously adopted on the Geroch–Kronheimer–Penrose causal completion (or c-completion for short) of . The CLT retains essentially all the good properties of other topologies previously defined in the literature, such as the well-known chronological topology. In this paper we prove that if a globally hyperbolic spacetime admits a conformal boundary, defined in such broad terms as to include all the standard examples in the literature, then the latter is homeomorphic to the causal boundary endowed with the CLT. Then, we revisit the new definitions of (future) null infinity and black hole as given in terms of the c-boundaries in another recent work Costa e Silva et al (2018 J. High Energy Phys. JHEP(2018)123). We show that a number of the attending technicalities can be simplified for globally hyperbolic spacetimes when the CLT is considered.

Ricci Flow and Volume Renormalizability

With respect to any special boundary defining function, a conformally compact asymptotically hyperbolic metric has an asymptotic expansion near its conformal infinity. If this expansion is even to a certain order and satisfies one extra condition, then it is possible to define its renormalized volume and show that it is independent of choices that preserve this evenness structure. We prove that such expansions are preserved under normalized Ricci flow. We also study the variation of curvature functionals in this setting, and as one application, obtain the variation formula $$ \frac{{\rm d}}{{\rm d}t} {\rm RenV}\big(M^n, g(t)\big) = -\mathop{\vphantom{T}}^R \! \! \! \int_{M^n} (S(g(t))+n(n-1)) {\rm d}V_{g(t)},$$ where $S(g(t))$ is the scalar curvature for the evolving metric $g(t)$, and $\mathop{\vphantom{T}}^R \! \! \! \int (\cdot) {\rm d}V_g$ is Riesz renormalization. This extends our earlier work to a broader class of metrics.

High-efficiency X-ray multilayer-coated blazed gratings with shifted boundaries.

A new design for a high-efficiency multilayer-coated blazed X-ray grating with horizontal-shifted (non-conformal) boundary profiles is proposed. The investigation of the grating design is carried out using an integrated approach based on rigorous numerical calculations of light diffraction by gratings with realistic boundary profiles obtained from simulations of multilayer grating growth. By varying the incidence angle of the deposition flux, one can set the direction and magnitude of the boundary profile shifts over a wide range of values. It is shown that the diffraction efficiency of the blazed gratings with shifted boundary profiles may be substantially higher than the efficiency of gratings with conformal boundaries, which are, moreover, much more difficult to produce. High-efficiency gratings with shifted boundaries can be obtained when the deposition is mainly on the blaze facet with a high inclination of the deposition flux, as opposed to widely used near-normal deposition methods. The maximum absolute efficiency of a W/B4C 2500 mm-1 grating with a blaze angle of 1.76° and an anti-blaze angle of 20°, working at a blaze wavelength of 1.3 nm and having shifted realistic boundary profiles, obtained using our integrated approach is 23.3%, while that of a grating with the ideal (triangular) boundary profile and the same shifts is 25.3%, and that of an ideal conformal profile is only 22.2%. The maximum absolute efficiency of 40.2% of a 2500 mm Cr/C grating with a blaze angle of 1.05° and a realistic anti-blaze angle of 10°, working at a blaze wavelength of 0.83 nm and having ideal shifted boundaries, is higher than the maximum efficiency of the similar grating having ideal conformal boundaries with a non-realistic anti-blaze angle of 80°.

Comments on holographic entanglement entropy in TT deformed conformal field theories

The Ryu-Takayanagi (RT) formula has been a key ingredient in our understanding of holography. Recent work on TT deformations has also boosted our understanding of holography away from the conformal boundary of AdS. In this short note, we aim to refine some recent work demonstrating the success of the RT formula in TT deformed theories. We emphasize general arguments that justify the use of the RT formula in general holographic theories that obey a GKPW-like dictionary. In doing so, we clarify subtleties related to holographic counterterms and discuss the implications for holography in general spacetimes.

Black hole microstate cosmology

In this note, we explore the possibility that certain high-energy holographic CFT states correspond to black hole microstates with a geometrical behind-the-horizon region, modelled by a portion of a second asymptotic region terminating at an end-of-the-world (ETW) brane. We study the time-dependent physics of this behind-the-horizon region, whose ETW boundary geometry takes the form of a closed FRW spacetime. We show that in many cases, this behind-the-horizon physics can be probed directly by looking at the time dependence of entanglement entropy for sufficiently large spatial CFT subsystems. We study in particular states defined via Euclidean evolution from conformal boundary states and give specific predictions for the behavior of the entanglement entropy in this case. We perform analogous calculations for the SYK model and find qualitative agreement with our expectations. We also calculate holographic complexity for the d = 2 ETW geometries, finding that complexity-action and complexity-volume proposals give the same linear growth at late times, but differ at early times.A fascinating possibility is that for certain states, we might have gravity localized to the ETW brane as in the Randall-Sundrum II scenario for cosmology. In this case, the effective description of physics beyond the horizon could be a big bang/big crunch cosmology of the same dimensionality as the CFT. In this case, the d-dimensional CFT describing the black hole microstate would give a precise, microscopic description of the d-dimensional cosmological physics.

Lorentzian Einstein metrics with prescribed conformal infinity

We prove a local well-posedness theorem for the $(n+1)$-dimensional Einstein equations in Lorentzian signature, with initial data $(\widetilde{g},K)$ whose asymptotic geometry at infinity is similar to that anti-de Sitter (AdS) space, and compatible boundary data $\widehat{g}$ prescribed at the time-like conformal boundary of space-time. More precisely, we consider an $n$-dimensional asymptotically hyperbolic Riemannian manifold $(M, \widetilde{g})$ such that the conformally rescaled metric $x^2 \widetilde{g}$ (with $x$ a boundary defining function) extends to the closure $\overline{M}$ of $M$ as a metric of class $C^{n-1} (\overline{M})$ which is also poly-homogeneous of class $C^p_{\mathrm{polyhom}} (\overline{M})$. Likewise we assume that the conformally rescaled symmetric $(0, 2)$-tensor $x^ 2 K$ extends to $\overline{M}$ as a tensor field of class $C^{n-1} (\overline{M})$ which is polyhomogeneous of class $C^{p-1}_{\mathrm{polyhom}} (\overline{M})$. We assume that the initial data $(\widetilde{g}, K)$ satisfy the Einstein constraint equations and also that the boundary datum is of class $C^p$ on $\partial M \times (-T_0, T_0)$ and satisfies a set of natural compatibility conditions with the initial data. We then prove that there exists an integer $r_n$, depending only on the dimension $n$, such that if $p \geqslant 2q + r_n$, with $q$ a positive integer, then there is $T \gt 0$, depending only on the norms of the initial and boundary data, such that the Einstein equations (1.1) has a unique (up to a diffeomorphism) solution $g$ on $(-T, T) \times M$ with the above initial and boundary data, which is such that $x^2 g \in C^{n-1} ((-T, T) \times \overline{M}) \; \cap \; C^q_{\mathrm{polyhom}} ((-T, T) \times \overline{M})$. Furthermore, if $x^2 \widetilde{g} , x^2 K$ are polyhomogeneous of class $C^{\infty}$ and $\widehat{g}$ is in $C^{\infty} ((-T_0, T_0) \times \partial \overline{M})$, then $x^2 g$ is in $C^{\infty}_{\mathrm{polyhom}} ((-T, T) \times \overline{M})$.

Boundary conditions and renormalized stress-energy tensor on a Poincaré patch of AdS2

Quantum field theory on anti-de Sitter spacetime requires the introduction of boundary conditions at its conformal boundary, due essentially to the absence of global hyperbolicity. Here we calculate the renormalized stress-energy tensor $T_{\mu\nu}$ for a scalar field $\phi$ on the Poincar\'e patch of $\text{AdS}_2$ and study how it depends on those boundary conditions. We show that, except for the Dirichlet and Neumann cases, the boundary conditions break the maximal $\textrm{AdS}$ invariance. As a result, $\langle\phi^2\rangle$ acquires a space dependence and $\langle T_{\mu\nu}\rangle$ is no longer proportional to the metric. When the physical quantities are expanded in a parameter $\beta$ which characterizes the boundary conditions (with $\beta=0$ corresponding to Dirichlet and $\beta=\infty$ corresponding to Neumann), the singularity of the Green's function is entirely subtracted at zeroth order in $\beta$. As a result, the contribution of nontrivial boundary conditions to the stress-energy tensor is free of singular terms.

Comment on “Hadamard states for a scalar field in anti–de Sitter spacetime with arbitrary boundary conditions”

In a recent paper (Phys. Rev. D 94, 125016 (2016)), the authors argued that the singularities of the two-point functions on the Poincar\'e domain of the $n$-dimensional anti-de Sitter spacetime ($\text{PAdS}_n$) have the Hadamard form, regardless of which (Robin) boundary condition is chosen at the conformal boundary. However, the argument used to prove this statement was based on an incorrect expression for the two-point function $G^{+}(x,x')$, which was obtained by demanding $\text{AdS}$ invariance for the vacuum state. In this comment I show that their argument works only for Dirichlet and Neumann boundary conditions and that the full $\text{AdS}$ symmetry cannot be respected by nontrivial Robin conditions (i.e., those which are neither Dirichlet nor Neumann). By studying the conformal scalar field on $\text{PAdS}_2$, I find the correct expression for $G^{+}(x,x')$ and show that, notwithstanding this problem, it still have the Hadamard form.

A construction of Poincaré–Einstein metrics of cohomogeneity one on the ball

We exhibit an explicit one-parameter smooth family of Poincaré–Einstein metrics on the even-dimensional unit ball whose conformal infinities are the Berger spheres. Our construction is based on a Gibbons–Hawking-type ansätz of Page and Pope. The family contains the hyperbolic metric, converges to the complex hyperbolic metric at one of the ends, and at the other end the ball equipped with our metric collapses to a Poincaré–Einstein manifold of one lower dimension with an isolated conical singularity.

(A)dS4 in Bondi gauge

We obtain the general asymptotic solutions of Einstein gravity with or without cosmological constant in Bondi gauge. The Bondi gauge was originally introduced in the context of gravitational radiation in asymptotically flat gravity. In the original work, initial conditions were prescribed at a null hypersurface and the Einstein equations were shown to take a nested form, which may be used to explicitly integrate them asymptotically. We streamline the derivation of the general asymptotic solution in the asymptotically flat case, and derive the most general asymptotic solutions for the case of non-zero cosmological constant of either sign (asymptotically locally AdS and dS solutions). With non-zero cosmological constant, we present integration schemes which rely on either prescribing data on the conformal boundary or on a null hypersurface and part of the conformal boundary. We explicitly work out the transformation to Fefferman–Graham gauge and identity how to extract the holographic data directly in Bondi coordinates. We illustrate the discussion with a number of examples and show that for asymptotically AdS4 spacetimes the Bondi mass is constant.

A mechanism for holography for non-interacting fields on anti-de Sitter spacetimes

In the setting of non-interacting Klein–Gordon fields on asymptotically anti-de Sitter spacetimes, we show that algebras of observables localized in a neighborhood of the boundary are subalgebras of a boundary algebra. The underlying mechanism is directly related to holography for classical fields. In particular, the proof relies on unique continuation theorems at the conformal boundary.

Lyapunov-Type Inequalities for a Conformable Fractional Boundary Value Problem of Order 3<α≤4

We establish new Lyapunov-type inequalities for the following conformable fractional boundary value problem (BVP): Tαaut+q(t)u(t)=0, a<t<b, u(a)=u′(a)=u′′(a)=u′′(b)=0, where Tαa is the conformable fractional derivative of order α∈(3,4] and q is a real-valued continuous function. Some applications to the corresponding eigenvalue problem are discussed.

On the Poisson boundary of the relativistic Brownian motion

Dans cet article, nous determinons la frontiere de Poisson du mouvement brownien relativiste dans deux classes de varietes lorentziennes, les espaces modeles de courbure constante et les espaces de Robertson–Walker qui constituent une vaste famille d’espace-temps courbes. Notre objectif est double : d’une part, il s’agit de comprendre les relations entre la geometrie a l’infini de ces varietes et le comportement asymptotique des trajectoires browniennes, avec comme objectif de comparer la compactification stochastique formee par les points de sortie du processus aux frontieres purement geometriques, conformes ou causales. D’autre part, nous souhaitons illustrer la pertinence de la methode de devissage introduite par les auteurs (Seminaire de Probabilites XLVIII (2016) 199–229 Springer), methode qui s’avere particulierement bien adaptee aux differents contextes geometriques consideres.

Conformal extension of the Bunch-Davies state across the de Sitter boundary

In the setting of the massive Klein-Gordon equation on de Sitter space, we discuss Vasy’s asymptotic data at conformal infinity in terms of plane waves. In particular, we derive a short-hand formula for reconstructing solutions from their asymptotic data. Furthermore, we show that the natural Hadamard state induced from future (or past) conformal infinity coincides with the Bunch-Davies state.

Variational calculus for hypersurface functionals: Singular Yamabe problem Willmore energies

We develop an efficient calculus for varying hypersurface embeddings based on variations of hypersurface defining functions. This is used to show that the functional gradient of a new Willmore-like, conformal hypersurface energy agrees exactly with the obstruction to smoothly solving the singular Yamabe problem for conformally compact four-manifolds. We give explicit formulæ for both the energy functional and the obstruction. Vanishing of the latter is a necessary condition for solving the vacuum cosmological Einstein equations in four spacetime dimensions with data prescribed on a conformal infinity, while the energy functional generalizes the scheme-independent contribution to entanglement entropy across surfaces to hypersurfaces.

Holographic entanglement entropy for black strings

The holographic entanglement entropy (HEE) of field theories dual to 4-dimensional black strings with both the infinity boundary and near the horizon is investigated. It is shown that, with the infinity boundary, the main contribution of the HEE is divergent and it comes from the AdS background. If we remove this contribution, the remaining part is finite and proportional to the mass and charge densities. Near the horizon, we find that the HEE equals the Bekenstein–Hawking entropy if the subsystem covers the whole conformal boundary, for both charged and uncharged cases. In particular, the minimal surface (Ryu–Takayanagi surface) always stays in the bulk region for uncharged and non-extremal charged cases, while it is immersed inside the boundary for extremal black string.

A New Understanding of the Lower Cretaceous Jiufotang Formation in Western Liaoning

The Lower Cretaceous Jiufotang Formation in western Liaoning is the most important fossil production horizon of the Jehol Biota, which is widely distributed in the Mesozoic basins of western Liaoning. Due to the influence of historical data, previous scholars believed that there was no volcanic activity in the Jiufotang Formation in western Liaoning. In a field investigation in western Liaoning, the authors discovered basalt and andesite in the Hujiayingzi bed. In addition, a conformable boundary was found between the Yixian and the Jiufotang formations. It indicates that both the Jiufotang Formation and the Yixian Formation are strata containing volcanic-sedimentary rocks, only differing in strength of volcanic activity.

Computing the Dirichlet--Neumann Operator on a Cylinder

The computation of the Dirichlet--Neumann operator for the Laplace equation is the primary challenge for the numerical simulation of the ideal fluid equations. The techniques commonly used for 2D fluids, such as conformal mapping and boundary integral methods, fail to generalize suitably to three dimensions. In this study, we address this problem by developing a Transformed Field Expansion method for computing the Dirichlet--Neumann operator in a cylindrical geometry with a variable upper boundary. This technique reduces the problem to a sequence of Poisson equations on a flat geometry. We design a fast and accurate solver for these subproblems, a key ingredient of which is the use of Zernike polynomials for the circular cross-section instead of the traditional Bessel functions. This lends spectral accuracy to the method and allowing significant computational speed-up. We rigorously analyze the algorithm, prove its applicability to a wide class of problems, and demonstrate its effectiveness numerically.

Chiral entanglement in massive quantum field theories in 1+1 dimensions

We determine both analytically and numerically the entanglement between chiral degrees of freedom in the ground state of massive perturbations of 1+1 dimensional conformal field theories quantised on a cylinder. Analytic predictions are obtained from a variational Ansatz for the ground state in terms of smeared conformal boundary states recently proposed by J. Cardy, which is validated by numerical results from the Truncated Conformal Space Approach. We also extend the scope of the Ansatz by resolving ground state degeneracies exploiting the operator product expansion. The chiral entanglement entropy is computed both analytically and numerically as a function of the volume. The excellent agreement between the analytic and numerical results provides further validation for Cardy’s Ansatz. The chiral entanglement entropy contains a universal O(1) term γ for which an exact analytic result is obtained, and which can distinguish energetically degenerate ground states of gapped systems in 1+1 dimensions.