Anti-de Sitter Geometry Research Articles

Classical boundary-value problem in Riemannian quantum gravity and Taub–Bolt–anti-de Sitter geometries

For an SU(2)× U(1)-invariant S 3 boundary the classical Dirichlet problem of Riemannian quantum gravity is studied for positive-definite regular solutions of the Einstein equations with a negative cosmological constant within biaxial Bianchi-IX metrics containing bolts, i.e., within the family of Taub–Bolt–anti-de Sitter (Taub–Bolt–AdS) metrics. Such metrics are obtained from the two-parameter Taub–NUT–anti-de Sitter family. The condition of regularity requires them to have only one free parameter ( L) and constrains L to take values within a narrow range; the other parameter is determined as a double-valued function of L and hence there is a bifurcation within the family. We found that any axially symmetric S 3-boundary can be filled in with at least one solution coming from each of these two branches despite the severe limit on the permissible values of L. The number of infilling solutions can be one, three or five and they appear or disappear catastrophically in pairs as the values of the two radii of S 3 are varied. The solutions occur simultaneously in both branches and hence the total number of independent infillings is two, six or ten. We further showed that when the two radii are of the same order and large the number of solutions is two. In the isotropic limit this holds for small radii as well. These results are to be contrasted with the one-parameter self-dual Taub–NUT–AdS infilling solutions of the same boundary-value problem studied previously.

Duality of boundary value problems and braneworld action in curved brane models

Braneworld effective action is constructed by two different methods based respectively on the Dirichlet and Neumann boundary value problems. The equivalence of these methods is shown due to non-trivial duality relations between special boundary operators of these two problems. Previously known braneworld action algorithms in two-brane Randall–Sundrum model are generalized to d-dimensional curved branes with de Sitter and anti-de Sitter geometries. Implications of the obtained algorithms in braneworld scenarios and in the cosmological constant and cosmological acceleration problems are briefly discussed.

Holonomies, anomalies and the Fefferman–Graham ambiguity in gravity

Using the Chern-Simon formulation of (2+1) gravity, we derive, for the general asymptotic metrics given by the Fefferman-Graham-Lee theorems, the emergence of the Liouville mode associated to the boundary degrees of freedom of (2+1) dimensional anti de Sitter geometries. Holonomies are described through multi-valued gauge and Liouville fields and are found to algebraically couple the fields defined on the disconnected components of spatial infinity. In the case of flat boundary metrics, explicit expressions are obtained for the fields and holonomies. We also show the link between the variation under diffeomorphisms of the Einstein theory of gravitation and the Weyl anomaly of the conformal theory at infinity.

Thick domain walls and singular spaces

We discuss thick domain walls interpolating between spaces with naked singularities and give arguments based on the $AdS$/CFT correspondence why such singularities may be physically meaningful. Our examples include thick domain walls with Minkowski, de Sitter, and anti-de Sitter geometries on the four-dimensional slice. For flat domain walls we can solve the equivalent quantum mechanics problem exactly, which provides the spectrum of graviton states. In one of the examples we discuss, the continuum states have a mass gap. We compare the graviton spectra with expectations from the $AdS$/CFT correspondence and find qualitative agreement. We also discuss unitary boundary conditions and show that they project out all continuum states.

The shape of gravity

In a non-trivial background geometry with extra dimensions, gravitational effects will depend on the shape of the Kaluza-Klein excitations of the graviton. We investigate a consistent scenario of this type with two positive tension three-branes separated in a five-dimensional anti-de Sitter geometry. The graviton is localized on the ``Planck'' brane, while a gapless continuum of additional gravity eigenmodes probe theinfinitely large fifth dimension. Despite the background five-dimensional geometry, an observer confined to either brane sees gravity as essentially four-dimensional up to a position-dependent strong coupling scale, no matter where the brane is located. We apply this scenario to generate the TeV scale as a hierarchically suppressed mass scale. Arbitrarily light gravitational modes appear in this scenario, but with suppressed couplings. Real emission of these modes is observable at future colliders; the effects are similar to those produced by six large toroidal dimensions.

Exponential and power-law hierarchies from supergravity

We examine how a d-dimensional mass hierarchy can be generated from a d+1-dimensional set up. We introduce a d+1-dimensional scalar, the hierarchon, which has a potential as in gauged supergravities. We find that when it is in its minimum, there exist solutions of Hořava-Witten topology R d×S 1/ Z 2 with domain walls at the fixed points and anti-de Sitter geometry in the bulk. We show that while standard Poincaré supergravity leads to power-law hierarchies, (e.g. a power law dependence of masses on the compactification scale), gauged supergravity produce an exponential hierarchy as recently proposed by Randall and Sundrum.

Holographic description of D3-branes in flat space

We describe a scheme for constructing the holographic dual of the full D3-brane geometry with charge $K$ by embedding it into a large anti-de Sitter space of size $N$. Such a geometry is realized in a multi-center anti-de Sitter geometry which admits a simple field theory interpretation as $SU(N+K)$ gauge theory broken to $SU(N) \times SU(K)$. We find that the characteristic size of the D3-brane geometry is of order $(K/N)^{1/4} U^0$ where $U^0$ is the scale of the Higgs. By choosing $N$ to be much larger than $K$, the scale of the D3-brane metric can be well separated from the Higgs scale in the radial coordinate. We generalize the holographic energy-distance relation and estimate the characteristic energy scale associated with these radial scales, and find that the $E/U$ relation becomes effectively $U$ independent in the range $(K/N)^{1/2} U^0 < U < U^0$. This implies that all detailed structure of the D3-brane geometry is encoded in the fine structure of the boundary gauge theory at around the Higgs scale.

Gödel metric as a squashed anti-de Sitter geometry

We show that the non-flat factor of the Gödel metric belongs to a one-parameter family of (2 + 1)-dimensional geometries that also includes the anti-de Sitter metric. The elements of this family allow a generalization à la Kaluza-Klein of the usual (3 + 1)-dimensional Gödel metric. Their lightcones can be viewed as deformations of the anti-de Sitter ones, involving tilting and squashing. This provides a simple geometric picture of the causal structure of these spacetimes, anti-de Sitter geometry appearing as the boundary between causally safe and causally pathological spaces. Furthermore, we construct a global algebraic isometric embedding of these metrics in (4 + 3)- or (3 + 4)-dimensional flat spaces, thereby illustrating in another way the occurrence of the closed timelike curves.

Anti de Sitter geometry and strongly coupled gauge theories

We propose supergravity duals to non-trivial fixed points of the renormalization group in three dimensions with ADE global symmetries. All the fixed point symmetries are identified with space-time symmetries.