Supersymmetry in Anti-de Sitter Space

We review the renormalization of the ground state solution of extended supergravity and super-symmetric Kaluza-Klein theories. The computation of an adiabatic expansion of the effective action to the one-loop order yields the result that a linear superfield insertion in the superpotential is needed, in order to renormalize the nonvanishing one-particle-irreducible one-point functions, whereas supersymmetry is preserved at each extremum of the effective potential. The calculation of the one-particle-irreducible two-and three-point functions shows that neither the mass nor the interaction lagrangians get renormalized to the one-loop order. We conclude that the one-loop effects proportional to the contraction parameter of the curved background space force a violation of the no-renormalization theorem.     

Collapse of Self-Interacting Scalar Field in Anti-de Sitter Space

The gravitational collapse of a massless scalar field with a self-interaction term λφ~4 in anti-de Sitter space is investigated.We numerically investigate the effect of the self-interaction term on the critical amplitudes,forming time of apparent horizon,stable island,and energy transformation.The results show that a positiveλsuppresses the formation of black hole,while a negativeλenhances the process.We define two susceptibilities to characterize the effect of the self-interaction on the black hole formation,and find that near the critical amplitude,there exists a universal scaling relation with the critical exponentα≈0.74 for the time of black hole formation.     

A Static Solution of Yang-Mills Equation on Anti-de Sitter Space

Since product metric on AdS space has played a very important role in Lorentz version of AdS/CFT correspondence, the Yang Mills equation on AdS space with this metric is considered and a static solution is obtained in this paper, which helps to understand the AdS/CFT correspondence of Yang Mills fields.     

Neutrino Oscillations in the de Sitter and the Anti-de Sitter Space-Time

General expressions of the neutrino oscillation phase in the generally static space-time with spherical symmetry are given. The effect of the gravitational field on the oscillation length is embodied in the gravitational red shift factor. We find that a blue shift of the oscillation length takes place when the neutrino travels out of the gravitational field. Then, we discuss the variation of the oscillation length influenced by the cosmological constant. In the de Sitter space-time, the positive cosmological constant prolongs the oscillation length. And, in the anti-de Sitter space-time, the negative cosmological constant shortens it as expected.     

Anti-de Sitter space at finite temperature

We consider a conformally invariant scalar field at finite temperature in anti-de Sitter space, and find the symmetric two-point function. Since it is meromorphic and it has both a real-time and imaginary-time periodicity, it is an elliptic function. From it, the expectation values of ø² and the stress-energy tensor are calculated exactly, and then compared to a Tolman-redshifted radiation gas, and to Page's “optical” approximation. The total energy of the radiation is finite.     

Exact Non-Singular Waves in the Anti-de Sitter Universe

A class of radiative solutions of Einstein's field equations with a negative cosmological constant and a pure radiation is investigated. The space-times, which generalize the Defrise solution, represent exact gravitational waves which interact with null matter and propagate in the anti–de Sitter universe. Interestingly, these solutions have homogeneous and non-singular wave-fronts for all freely moving observers. We also study properties of sandwich and impulsive waves which can be constructed in this class of space-times.     

Unique Continuation from Infinity in Asymptotically Anti-de Sitter Spacetimes

We consider the unique continuation properties of asymptotically anti-de Sitter spacetimes by studying Klein–Gordon-type equations \({\Box_g \phi + \sigma \phi = {\mathcal{G}} ( \phi, \partial \phi )}\), \({\sigma \in {\mathbb{R}}}\), on a large class of such spacetimes. Our main result establishes that if \({\phi}\) vanishes to sufficiently high order (depending on \({\sigma}\)) on a sufficiently long time interval along the conformal boundary \({{\mathcal{I}}}\), then the solution necessarily vanishes in a neighborhood of \({{\mathcal{I}}}\). In particular, in the \({\sigma}\)-range where Dirichlet and Neumann conditions are possible on \({{\mathcal{I}}}\) for the forward problem, we prove uniqueness if both these conditions are imposed. The length of the time interval can be related to the refocusing time of null geodesics on these backgrounds and is expected to be sharp. Some global applications as well as a uniqueness result for gravitational perturbations are also discussed. The proof is based on novel Carleman estimates established in this setting.     

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     

Parallel and Killing Spinors on Spinc Manifolds

We describe all simply connected Spin^c manifolds carrying parallel and real Killing spinors. In particular we show that every Sasakian manifold (not necessarily Einstein) carries a canonical Spin^c structure with Killing spinors. Received: 24 December 1996 / Accepted: 6 January 1997     

Analytical Solutions of the Gravitational Field Equations in de Sitter and Anti-de Sitter Spacetimes

The generalized Laplace partial differential equation, describing gravitational fields, is investigated in de Sitter spacetime from several metric approaches—such as the Riemann, Beltrami, Börner-Dürr, and Prasad metrics—and analytical solutions of the derived Riccati radial differential equations are explicitly obtained. All angular differential equations trivially have solutions given by the spherical harmonics and all radial differential equations can be written as Riccati ordinary differential equations, which analytical solutions involve hypergeometric and Bessel functions. In particular, the radial differential equations predict the behavior of the gravitational field in de Sitter and anti-de Sitter spacetimes, and can shed new light on the investigations of quasinormal modes of perturbations of electromagnetic and gravitational fields in black hole neighborhood. The discussion concerning the geometry of de Sitter and anti-de Sitter spacetimes is not complete without mentioning how the wave equation behaves on such a background. It will prove convenient to begin with a discussion of the Laplace equation on hyperbolic space, partly since this is of interest in itself and also because the wave equation can be investigated by means of an analytic continuation from the hyperbolic space. We also solve the Laplace equation associated to the Prasad metric. After introducing the so called internal and external spaces—corresponding to the symmetry groups SO(3,2) and SO(4,1) respectively—we show that both radial differential equations can be led to Riccati ordinary differential equations, which solutions are given in terms of associated Legendre functions. For the Prasad metric with the radius of the universe independent of the parametrization, the internal and external metrics are shown to be of AdS-Schwarzschild-like type, and also the radial field equations arising are shown to be equivalent to Riccati equations whose solutions can be written in terms of generalized Laguerre polynomials and hypergeometric confluent functions.     

Thermodynamics of DBI Black Holes in Anti-de Sitter Spacetime

Through the gauge field theory, we obtain the solution of the DBI-AdS black hole. In the meantime, according to the relations between the action and the grand partition function, we obtain the grand partition function in the DBI-AdS black hole. The temperature and the potential of the DBI-AdS black hole are gained from differential of the grand partition function. With the thermodynamic relations, other thermodynamics are also obtained. The solution and the thermodynamics of the DBI-AdS black hole are turned out that they can reduce to the case of a charged black hole in four-dimensional spacetimes.     

The Dirac System on the Anti-de Sitter Universe

We investigate the global solutions of the Dirac equation on the Anti- de-Sitter Universe. Since this space is not globally hyperbolic, the Cauchy problem is not, a priori, well-posed. Nevertheless we can prove that there exists unitary dynamics, but its uniqueness crucially depends on the ratio beween the mass M of the field and the cosmological constant Λ > 0: it appears a critical value, Λ/12, which plays a role similar to the Breitenlohner-Freedman bound for the scalar fields. When M ² ≥  Λ/12 there exists a unique unitary dynamics. On the contrary, for the light fermions satisfying M ² < Λ/12, we construct several asymptotic conditions at infinity, such that the problem becomes well-posed. In all the cases, the spectrum of the hamiltonian is discrete. We also prove a result of equipartition of the energy.     

New Dynamics in the Anti-de Sitter Universe AdS 5

This paper deals with the propagation of the gravitational waves in the Poincaré patch of the 5-dimensional Anti-de Sitter universe. We construct a large family of unitary dynamics with respect to some high order energies that are conserved and positive. These dynamics are associated with asymptotic conditions on the conformal time-like boundary of the universe. This result does not contradict the statement of Breitenlohner-Freedman that the hamiltonian is essentially self-adjoint in L ² and thus accordingly the dynamics is uniquely determined. The key point is the introduction of a new Hilbert functional framework that contains the massless graviton which is not normalizable in L ². Then the hamiltonian is not essentially self-adjoint in this new space and possesses a lot of different positive self-adjoint extensions. These dynamics satisfy a holographic principle: there exists a renormalized boundary value which completely characterizes the whole field in the bulk.     

Anti-de Sitter时空内柱黑洞的量子熵

利用brick-wall方法计算了Anti-de Sitter时空内起源于Dirac场的柱黑洞的量子熵.结果表明，忽略远离围绕系统的真空的贡献时，量子熵包含了线性发散项和对数发散项，整个表达式的形式与标量场的不一样.无论整个对数项还是与自旋联系的子对数项都总是正的. 关键词： brick-wall方法 量子熵 柱黑洞 Dirac场     

The Gauss map of minimal surfaces in the Anti-de Sitter space

It is proved that a pair of spinors satisfying a Dirac-type equation represent surfaces immersed in Anti-de Sitter space with prescribed mean curvature. Here, we consider Anti-de Sitter space as the Lie group SU_1,1${\text{<small-caps>SU</small-caps>}}_{1,1}$ endowed with a one-parameter family of left-invariant metrics where only one of them is bi-invariant and corresponds to the isometric embedding of Anti-de Sitter space as a quadric in R^2,2${\mathbb{R}}^{2,2}$. We prove that the Gauss map of a minimal surface immersed in SU_1,1${\text{<small-caps>SU</small-caps>}}_{1,1}$ is harmonic. Conversely, we exhibit a representation of minimal surfaces in Anti-de Sitter space in terms of a given harmonic map.     

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.     

On 3+1 Anti-de Sitter and de Sitter Lie Bialgebras with Dimensionful Deformation Parameters

We analyze among all possible quantum deformations of the 3+1 (anti)de Sitter algebras, so(3,2) and so(4,1), which have two specific non-deformed or primitive commuting operators: the time translation/energy generator and a rotation. We prove that under these conditions there are only two families of two-parametric (anti)de Sitter Lie bialgebras. All the deformation parameters appearing in the bialgebras are dimensionful ones and they may be related to the Planck length. Some properties conveyed by the corresponding quantum deformations (zero-curvature and non-relativistic limits, space isotropy, . . . ) are studied and their dual (first-order) non-commutative spacetimes are also presented.