Holography at an extremal De Sitter horizon

Rotating maximal black holes in four-dimensional de Sitter space, for which the outer event horizon coincides with the cosmological horizon, have an infinite near-horizon region described by the rotating Nariai metric. We show that the asymptotic symmetry group at the spacelike future boundary of the near-horizon region contains a Virasoro algebra with a real, positive central charge. This is evidence that quantum gravity in a rotating Nariai background is dual to a two-dimensional Euclidean conformal field theory. These results are related to the Kerr/CFT correspondence for extremal black holes, but have two key differences: one of the black hole event horizons has been traded for the cosmological horizon, and the near-horizon geometry is a fiber over dS₂ rather than AdS₂.     

Inflation and conformal invariance: the perspective from radial quantization

According to the dS/CFT correspondence, correlators of fields generated during a primordial de Sitter phase are constrained by three‐dimensional conformal invariance. Using the properties of radially quantized conformal field theories and the operator‐state correspondence, we glean information on some points. The Higuchi bound on the masses of spin‐s states in de Sitter is a direct consequence of reflection positivity in radially quantized CFT₃ and the fact that scaling dimensions of operators are energies of states. The partial massless states appearing in de Sitter correspond from the boundary CFT₃ perspective to boundary states with highest weight for the conformal group. Finally, we discuss the inflationary consistency relations and the role of asymptotic symmetries which transform asymptotic vacua to new physically inequivalent vacua by generating long perturbation modes. We show that on the CFT₃ side, asymptotic symmetries have a nice quantum mechanics interpretation. For instance, acting with the asymptotic dilation symmetry corresponds to evolving states forward (or backward) in “time” and the charge generating the asymptotic symmetry transformation is the Hamiltonian itself.     

Thermodynamics in rotating anti-de Sitter black holes with massive scalar field in three dimensions

We investigated the tendency in the variations of CFT₂ when a rotating AdS₃ black hole changes because of the fluxes transferred by the scattering of a massive scalar field according to the anti-de Sitter (AdS)/conformal field theory (CFT) correspondence. The conserved quantities of the black hole are definitely constrained by the extremal condition. Moreover, the laws of thermodynamics provide a direction for the changes in the conserved quantities. Therefore, the black hole cannot be extremal under the scattering; this is naturally preferred. According to the relationship between the rotating AdS₃ black hole and dual CFT₂, we find that such changes in the black hole constrain the variations in the eigenstates of dual CFT₂. Furthermore, the tendency in the variations is closely related to the laws of thermodynamics.     

Hadron potentials within the gauge/string correspondence

It is known, since the 70s, that the large N 't Hooft limit of gauge theories is related to string theories. In 1998, J. M. Maldacena identified precisely such a relation: the so-called AdS/CFT correspondence which speculates a duality between a large N strongly-coupled supersymmetric and conformal Yang-Mills theory in four dimensions and a weakly-coupled string theory defined in a five-dimensional anti-de Sitter AdS₅ space-time. This review aims at introducing concepts and methods used to derive, in the framework of the gauge/string correspondence, the interaction potentials of mesons and baryons at zero and finite temperature. The dual string configurations associated with the different kinds of hadrons are described and their behaviours at short and large distances are understood. Although the application of Maldacena's AdS/CFT conjecture to QCD is not straightforward, QCD being neither supersymmetric nor conformal, the AdS/QCD correspondence approach attempts to identify the dual theory of QCD. Especially, the study of heavy quark-antiquark bound-states leads to establish general dual criteria for the confinement.     

A Stress Tensor for Anti-de Sitter Gravity

We propose a procedure for computing the boundary stress tensor associated with a gravitating system in asymptotically anti-de Sitter space. Our definition is free of ambiguities encountered by previous attempts, and correctly reproduces the masses and angular momenta of various spacetimes. Via the AdS/CFT correspondence, our classical result is interpretable as the expectation value of the stress tensor in a quantum conformal field theory. We demonstrate that the conformal anomalies in two and four dimensions are recovered. The two dimensional stress tensor transforms with a Schwarzian derivative and the expected central charge. We also find a nonzero ground state energy for global AdS₅, and show that it exactly matches the Casimir energy of the dual super Yang–Mills theory on S ³×R. Received: 20 April 1999 / Accepted: 8 July 1999     

Hidden conformal symmetry of self-dual warped AdS<Subscript>3</Subscript> black holes in topological massive gravity

We extend the recently proposal of hidden conformal symmetry to the self-dual warped AdS₃ black holes in topological massive gravity. It is shown that the wave equation of massive scalar field with sufficient small angular momentum can be reproduced by the SL(2, R) Casimir quadratic operator. Due to the periodic identification in the φ direction, it is found that only the left section of hidden conformal symmetry is broken to U(1), while the right section is unbroken, which only gives the left temperature of dual CFT. As a check of the dual CFT conjecture of self-warped AdS₃ black hole, we further compute the Bekenstein–Hawking entropy and absorption cross section and quasinormal modes of scalar field perturbation and show these are just of the forms predicted by the dual CFT.     

Intertwined Hamiltonians in two-dimensional curved spaces

The problem of intertwined Hamiltonians in two-dimensional curved spaces is investigated. Explicit results are obtained for Euclidean plane, Minkowski plane, Poincaré half plane (AdS₂), de Sitter plane (dS₂), sphere, and torus. It is shown that the intertwining operator is related to the Killing vector fields and the isometry group of corresponding space. It is shown that the intertwined potentials are closely connected to the integral curves of the Killing vector fields. Two problems are considered as applications of the formalism presented in the paper. The first one is the problem of Hamiltonians with equispaced energy levels and the second one is the problem of Hamiltonians whose spectrum is like the spectrum of a free particle.     

The Cardy-Verlinde formula and entropyof topological Kerr- Newman black holes in de Sitter spaces

In this paper we show that the entropy of a cosmological horizon in 4-dimensional topological Kerr-Newman-de Sitter spaces can be described by the Cardy-Verlinde formula, which is supposed to be an entropy formula of conformal field theory in any number of dimensions. Furthermore, we find that the entropy of a black hole horizon can also be rewritten in terms of the Cardy-Verlinde formula for these black holes in de Sitter spaces, if we use the definition due to Abbott and Deser for conserved charges in asymptotically de Sitter spaces. Such results presume a well-defined dS/CFT correspondence, which has not yet attained the credibility of its AdS analogue.Received: 7 April 2003, Revised: 18 June 2003, Published online: 29 August 2003     

Anti-de Sitter BPS black holes in N=2 gauged supergravity

Electrically charged solutions breaking half of the supersymmetry in Anti-de Sitter four dimensional N=2 supergravity coupled to vector supermultiplets are constructed. These static black holes live in an asymptotic AdS₄ space time. The Killing spinor, i.e., the spinor for supersymmetry variation is explicitly constructed for these solutions.     

Unitarity and Page Curve for Evaporation of 2D AdS Black Holes

We explore the Hawking evaporation of two-dimensional anti-de Sitter (AdS2), dilatonic black hole coupled with conformal matter, and derive the Page curve for the entanglement entropy of radiation. We first work in a semiclassical approximation with backreaction. We show that the end-point of the evaporation process is AdS2 with a vanishing dilaton, i.e., a regular, singularity-free, zero-entropy state. We explicitly compute the entanglement entropies of the black hole and the radiation as functions of the horizon radius, using the conformal field theory (CFT) dual to AdS2 gravity. We use a simplified toy model, in which evaporation is described by the forming and growing of a negative mass configuration in the positive-mass black hole interior. This is similar to the “islands” proposal, recently put forward to explain the Page curve for evaporating black holes. The resulting Page curve for AdS2 black holes is in agreement with unitary evolution. The entanglement entropy of the radiation initially grows, closely following a thermal behavior, reaches a maximum at half-way of the evaporation process, and then goes down to zero, following the Bekenstein–Hawking entropy of the black hole. Consistency of our simplified model requires a non-trivial identification of the central charge of the CFT describing AdS2 gravity with the number of species of fields describing Hawking radiation.     

Near-horizon conformal symmetry and black hole entropy

Near an event horizon, the action of general relativity acquires a new asymptotic conformal symmetry. For two-dimensional dilaton gravity, this symmetry results in a chiral Virasoro algebra, and Cardy's formula for the density of states reproduces the Bekenstein-Hawking entropy. This lends support to the notion that black hole entropy is controlled universally by conformal symmetry near the horizon.     

Functional representation for the isometries of de Sitter space

We examine the Schrödinger picture for a spinless field theory in two-dimensional de Sitter space and construct an ultraviolet finite functional representation for the de Sitter Lie algebra. The one-parameter family of de Sitter vacua is found to be only phase-invariant, except for one value of the parameter where the state is truly invariant.     

Holographic dual of a boundary conformal field theory

We propose a holographic dual of a conformal field theory defined on a manifold with boundaries, i.e., boundary conformal field theory (BCFT). Our new holography, which may be called anti-de?Sitter BCFT, successfully calculates the boundary entropy or g function in two-dimensional BCFTs and it agrees with the finite part of the holographic entanglement entropy. Moreover, we can naturally derive a holographic g theorem. We also analyze the holographic dual of an interval at finite temperature and show that there is a first order phase transition.     

Hidden conformal symmetry and pair production near the cosmological horizon in Kerr-Newman-Taub-NUT-de Sitter spacetime

We show that the study of the hidden conformal symmetry that is associated with the Kerr/CFT correspondence can also apply to the cosmological horizon in the Kerr-Newman-Taub-NUT-de Sitter spacetime. This symmetry allows employing some two dimensional conformal field theory methods to understand the properties of the cosmological horizon. The entropy can be understood by using the Cardy formula, and the equation for the scattering process in the near region is in agreement with that obtained from a two point function in the two-dimensional conformal field theory. We also show that pair production can occur near the cosmological horizon in Kerr-Newman-Taub-NUT-de Sitter for near extremal conditions.     

Casimir Effect for Moving Branes in Static dS4+1 Bulk

In this paper we study the Casimir effect for conformally coupled massless scalar fields on background of Static dS₄₊₁ spacetime. We will consider the general plane–symmetric solutions of the gravitational field equations and boundary conditions of the Dirichlet type on the branes. Then we calculate the vacuum energy-momentum tensor in a configuration in which the boundary branes are moving by uniform proper acceleration in static de Sitter background. Static de Sitter space is conformally related to the Rindler space, as a result we can obtain vacuum expectation values of energy-momentum tensor for conformally invariant field in static de Sitter space from the corresponding Rindler counterpart by the conformal transformation.     

Auxiliary Conformally Invariant Higher-Spin Field in de Sitter Space

We employ de Sitter isometry to study a mixed symmetric rank-3 tensor field in de Sitter space by finding the field equation, solution and two-point function which are conformally invariant. It is proved that such a tensor field plays a key role in conformal theory of linear gravity (Binegar et al., Phys. Rev. D 27, 2249, 1983) . In de Sitter space from the group theoretical point of view this kind of tensor could associate with two unitary irreducible representations (UIR) of the de Sitter group (Takook et al., J.Math. Phys. 51, 032503, 2010), which one representation has a flat limit, namely, in zero curvature coincides to the UIR of Poincaré group, however, the second one which is named as auxiliary field, becomes significant in the study of conformal gravity in de Sitter background. We show that the rank-3 tensor solution can be written in terms of a massless minimally coupled scalar field and also the related two-point function is a function of a massless minimally coupled scalar two-point function.     

Thermodynamics of regular black hole

We investigate thermodynamics for a magnetically charged regular black hole (MCRBH), which comes from the action of general relativity and nonlinear electromagnetics, comparing with the Reissner–Norström (RN) black hole in both four and two dimensions after dimensional reduction. We find that there is no thermodynamic difference between the regular and RN black holes for a fixed charge Q in both dimensions. This means that the condition for either singularity or regularity at the origin of coordinate does not affect the thermodynamics of black hole. Furthermore, we describe the near-horizon AdS₂ thermodynamics of the MCRBH with the connection of the Jackiw–Teitelboim theory. We also identify the near-horizon entropy as the statistical entropy by using the AdS₂/CFT₁ correspondence.     

Conformal quantum mechanics as the CFT1 dual to AdS2

A (0+1)-dimensional candidate theory for the CFT₁ dual to AdS₂ is discussed. The quantum mechanical system does not have a ground state that is invariant under the three generators of the conformal group. Nevertheless, we show that there are operators in the theory that are not primary, but whose “non-primary character” conspires with the “non-invariance of the vacuum” to give precisely the correlation functions in a conformally invariant theory.