It was also discovered that pbranes, which are extended objects of higher dimension than strings (1branes), play a fundamental role in the theory. In the weak coupling limit, pbranes () become infinitely heavy, so that they do not appear in the perturbative theory. Of particular importance among pbranes are the Dbranes, on which open strings can end. Roughly speaking, open strings, which describe the nongravitational 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 [143], gauge fields of the standard model are confined on two 1+9branes 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 braneworld cosmology is given in [215, 216, 217].

These solutions can be thought of as effectively 5dimensional, with an extra dimension that can be large relative to the fundamental scale. They provide the basis for the Randall–Sundrum (RS) 2brane models of 5dimensional gravity [266] (see Figure 2). The singlebrane Randall–Sundrum models [265] 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) braneworld models [2, 9, 10, 11, 115, 119, 274, 313], which put forward the idea that a large volume for the compact extra dimensions would lower the fundamental Planck scale,
where is the electroweak scale. If is close to the lower limit in Equation (7), then this would address the longstanding “hierarchy” problem, i.e., why there is such a large gap between and .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 antide Sitter () spacetime. As in the Horava–Witten solutions, the RS branes are 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 selfgravity of the branes is incorporated in the RS models. The novel feature of the RS models compared to previous higherdimensional 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 braneworlds 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 RS2 models also provide a framework for exploring holographic ideas that have emerged in M theory. Roughly speaking, holography suggests that higherdimensional gravitational dynamics may be determined from knowledge of the fields on a lowerdimensional boundary. The AdS/CFT correspondence is an example, in which the classical dynamics of the higherdimensional gravitational field are equivalent to the quantum dynamics of a conformal field theory (CFT) on the boundary. The RS2 model with its metric satisfies this correspondence to lowest perturbative order [87] (see also [125, 136, 210, 254, 259, 282, 289, 293] for the AdS/CFT correspondence in a cosmological context).
In this review, I focus on RS braneworlds (mainly RS 1brane) and their generalizations, with the emphasis on geometry and gravitational dynamics (see [36, 79, 187, 188, 189, 190, 219, 228, 260, 267, 316] for previous reviews with a broadly similar approach). Other recent reviews focus on stringtheory aspects, e.g., [68, 101, 230, 264], or on particle physics aspects, e.g., [51, 104, 182, 262, 273]. Before turning to a more detailed analysis of RS braneworlds, I discuss the notion of Kaluza–Klein (KK) modes of the graviton.
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