It was also discovered that p-branes, which are extended objects of higher dimension than strings
(1-branes), play a fundamental role in the theory. In the weak coupling limit, p-branes () become
infinitely heavy, so that they do not appear in the perturbative theory. Of particular importance among
p-branes are the D-branes, on which open strings can end. Roughly speaking, open strings, which
describe the non-gravitational sector, are attached at their endpoints to branes, while the closed
strings of the gravitational sector can move freely in the bulk. Classically, this is realised via the
localization of matter and radiation fields on the brane, with gravity propagating in the bulk (see
Figure 1
).
In the Horava–Witten solution [203], gauge fields of the standard model are confined on two 1+9-branes
located at the end points of an orbifold, i.e., a circle folded on itself across a diameter. The 6 extra
dimensions on the branes are compactified on a very small scale close to the fundamental scale,
and their effect on the dynamics is felt through “moduli” fields, i.e., 5D scalar fields. A 5D
realization of the Horava–Witten theory and the corresponding brane-world cosmology is given
in [300, 301, 302].
These solutions can be thought of as effectively 5-dimensional, with an extra dimension that can be large
relative to the fundamental scale. They provide the basis for the Randall–Sundrum (RS) 2-brane models of
5-dimensional gravity [359] (see Figure 2
). The single-brane Randall–Sundrum models [358
]
with infinite extra dimension arise when the orbifold radius tends to infinity. The RS models
are not the only phenomenological realizations of M theory ideas. They were preceded by the
Arkani–Hamed–Dimopoulos–Dvali (ADD) brane-world models [15, 14, 13, 6, 367, 421, 168, 173], which
put forward the idea that a large volume for the compact extra dimensions would lower the fundamental
Planck scale,
In the ADD models, more than one extra dimension is required for agreement with experiments, and there is “democracy” amongst the equivalent extra dimensions, which are typically flat. By contrast, the RS models have a “preferred” extra dimension, with other extra dimensions treated as ignorable (i.e., stabilized except at energies near the fundamental scale). Furthermore, this extra dimension is curved or “warped” rather than flat: The bulk is a portion of anti-de Sitter (AdS5) spacetime. As in the Horava–Witten solutions, the RS branes are Z2-symmetric (mirror symmetry), and have a tension, which serves to counter the influence of the negative bulk cosmological constant on the brane. This also means that the self-gravity of the branes is incorporated in the RS models. The novel feature of the RS models compared to previous higher-dimensional models is that the observable 3 dimensions are protected from the large extra dimension (at low energies) by curvature rather than straightforward compactification.
The RS brane-worlds and their generalizations (to include matter on the brane, scalar fields in the bulk,
etc.) provide phenomenological models that reflect at least some of the features of M theory, and that
bring exciting new geometric and particle physics ideas into play. The RS models also provide a
framework for exploring holographic ideas that have emerged in M theory. Roughly speaking,
holography suggests that higher-dimensional gravitational dynamics may be determined from
knowledge of the fields on a lower-dimensional boundary. The AdS/CFT correspondence is
an example, in which the classical dynamics of the higher-dimensional gravitational field are
equivalent to the quantum dynamics of a conformal field theory (CFT) on the boundary. The RS
model with its AdS5 metric satisfies this correspondence to lowest perturbative order [129] (see
also [342
, 375
, 193
, 386
, 390
, 290
, 347
, 180
] for the AdS/CFT correspondence in a cosmological
context).
Before turning to a more detailed analysis of RS brane-worlds, We discuss the notion of Kaluza–Klein (KK) modes of the graviton.
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