Abstract:
The de Sitter manifold admits a wide variety of interesting coordinatizations. The 'atlas' is a compilation of the coordinate charts referenced throughout the literature, and is presented in the form of tables, the starting point being the embedding in a higher-dimensional Minkowski spacetime. The metric tensor and the references where the coordinate frame is discussed or used in applications are noted. Additional information is given for the entries with significant use: a convenient tetrad and the form taken by the Killing vectors in the respective coordinate frame.

Abstract:
We study the questions of how supersymmetry is spontaneously broken in Anti de-Sitter spacetime. We verify that the would-be R-symmetry in $AdS_4$ plays a central role for the existence of meta-stable supersymmetry breaking. To illustrate, some well-known models such as Poloyni models and O'Raifeartaigh models are investigated in detail. Our calculations are reliable in flat spacetime limit and confirm us that meta-stable vacua are generic even though quantum corrections are taken into account.

Abstract:
We prove that, among all (n + 1)-dimensional spin static vacua with positive cosmological constant, the de Sitter spacetime is characterized by the fact that its spatial Killing hori-zons have minimal modes for the Dirac operator. As a consequence, the de Sitter spacetime is the only vacuum of this type for which the induced metric tensor on some of its Killing horizons is at least equal to that of a round (n -- 1)-sphere. This extends unique-ness theorems shown by Boucher-Gibbons-Horowitz and Chruciel to more general horizon metrics and to the non-single horizon case.

Abstract:
In recent years, there has been considerable interest in theories formulated in anti-de Sitter (AdS) spacetime. However, AdS spacetime fails to be globally hyperbolic, so a classical field satisfying a hyperbolic wave equation on AdS spacetime need not have a well defined dynamics. Nevertheless, AdS spacetime is static, so the possible rules of dynamics for a field satisfying a linear wave equation are constrained by our previous general analysis--given in paper II--where it was shown that the possible choices of dynamics correspond to choices of positive, self-adjoint extensions of a certain differential operator, $A$. In the present paper, we reduce the analysis of electromagnetic, and gravitational perturbations in AdS spacetime to scalar wave equations. We then apply our general results to analyse the possible dynamics of scalar, electromagnetic, and gravitational perturbations in AdS spacetime. In AdS spacetime, the freedom (if any) in choosing self-adjoint extensions of $A$ corresponds to the freedom (if any) in choosing suitable boundary conditions at infinity, so our analysis determines all of the possible boundary conditions that can be imposed at infinity. In particular, we show that other boundary conditions besides the Dirichlet and Neumann conditions may be possible, depending on the value of the effective mass for scalar field perturbations, and depending on the number of spacetime dimensions and type of mode for electromagnetic and gravitational perturbations.

Abstract:
Classical geometry of de Sitter spacetime is reviewed in arbitrary dimensions. Topics include coordinate systems, geodesic motions, and Penrose diagrams with detailed calculations.

Abstract:
We study the Lamb shift of both freely-falling and static two-level atoms in interaction with quantized conformally coupled massless scalar fields in the de Sitter-invariant vacuum. We find that the Lamb shifts of both freely-falling and static atoms are in structural similarity to that of an inertial atom immersed in a thermal bath in a Minkowski spacetime. For the freely-falling atom, the Lamb shift gets a correction as if it was immersed in a thermal bath at the Gibbons-Hawking temperature, thus revealing clearly the intrinsic thermal nature of de Sitter spacetime. For the static atom, the Lamb shift is affected by a combination of the effect of the intrinsic thermal nature of de Sitter spacetime and the Unruh effect associated with the inherent acceleration of the atom.

Abstract:
We reformulate the Hamilton-Jacobi tunneling method for calculating Hawking radiation in static, spherically-symmetric spacetimes by explicitly incorporating a preferred family of frames. These frames correspond to a family of observers tied to a locally static timelike Killing vector of the spacetime. This formulation separates the role of the coordinates from the choice of vacuum and thus provides a coordinate-independent formulation of the tunneling method. In addition, it clarifies the nature of certain constants and their relation to these preferred observers in the calculation of horizon temperatures. We first use this formalism to obtain the expected temperature for a static observer at finite radius in the Schwarzschild spacetime. We then apply this formalism to the Schwarzschild-de Sitter spacetime, where there is no static observer with 4-velocity equal to the static timelike Killing vector. It is shown that a preferred static observer, one whose trajectory is geodesic, measures the lowest temperature from each horizon. Furthermore, this observer measures horizon temperatures corresponding to the well-known Bousso-Hawking normalization.

Abstract:
We consider, in de Sitter spacetime, both freely falling and static two-level atoms in interaction with a conformally coupled massless scalar field in the de Sitter-invariant vacuum, and separately calculate the contributions of vacuum fluctuations and radiation reaction to the atom's spontaneous excitation rate. We find that spontaneous excitations occur even for the freely falling atom as if there is a thermal bath of radiation at the Gibbons-Hawking temperature and we thus recover, in a different physical context, the results of Gibbons and Hawking that reveals the thermal nature of de Sitter spacetime. Similarly, for the case of the static atom, our results show that the atom also perceives a thermal bath which now arises as a result of the intrinsic thermal nature of de Sitter spacetime and the Unruh effect associated with the inherent acceleration of the atom.

Abstract:
We discuss the effective field theory for spacetime symmetry breaking from the local symmetry point of view. By gauging spacetime symmetries, the identification of Nambu-Goldstone (NG) fields and the construction of the effective action are performed based on the breaking pattern of diffeomorphism, local Lorentz, and (an)isotropic Weyl symmetries as well as the internal symmetries including possible central extensions in nonrelativistic systems. Such a local picture distinguishes, e.g., whether the symmetry breaking condensations have spins and provides a correct identification of the physical NG fields, while the standard coset construction based on global symmetry breaking does not. We illustrate that the local picture becomes important in particular when we take into account massive modes associated with symmetry breaking, whose masses are not necessarily high. We also revisit the coset construction for spacetime symmetry breaking. Based on the relation between the Maurer-Cartan one form and connections for spacetime symmetries, we classify the physical meanings of the inverse Higgs constraints by the coordinate dimension of broken symmetries. Inverse Higgs constraints for spacetime symmetries with a higher dimension remove the redundant NG fields, whereas those for dimensionless symmetries can be further classified by the local symmetry breaking pattern.

Abstract:
We show that the propagators of gravitons and scalar fields seen by a static patch observer in de Sitter spacetime are controlled by hidden SL(2,R) symmetries, at all frequencies. The retarded Green's function is determined by an SL(2,R) x SL(2,R) action generated by conformal Killing vectors of de Sitter spacetime times a line. This observation uses the fact that the static patch of dS_{d+1} x R is conformal to the hyperbolic patch of AdS_3 x S^{d-1}. The poles of the propagators, the quasinormal frequencies, are generated by associated SL(2,R) actions. The quasinormal mode generating algebras capture the conformal weights more usually read off from the fields at future and past infinity. For conformally coupled scalar fields, and for gravitons in four dimensions, this SL(2,R) algebra has an enhanced supersymmetric structure and is generated by particular conformal Killing vectors of de Sitter spacetime. We show how the worldline de Sitter propagators can be reproduced from a `level matched' left and right moving conformal quantum mechanics with an appropriate spectrum of primary operators. Our observations are consistent with the notion that the static patch of de Sitter spacetime is dually described by a (level matched) large N worldline conformal quantum mechanics.