Brane-worlds and M theory

1.2 Brane-worlds and M theory

String theory thus incorporates the possibility that the fundamental scale is much less than the Planck scale felt in 4 dimensions. There are five distinct 1+9-dimensional superstring theories, all giving quantum theories of gravity. Discoveries in the mid-1990s of duality transformations that relate these superstring theories and the 1+10-dimensional supergravity theory, led to the conjecture that all of these theories arise as different limits of a single theory, which has come to be known as M theory. The 11th dimension in M theory is related to the string coupling strength; the size of this dimension grows as the coupling becomes strong. At low energies, M theory can be approximated by 1+10-dimensional supergravity.

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 ( $p > 1$ ) 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 ).

Figure 1:

Schematic of confinement of matter to the brane, while gravity propagates in the bulk (from [51

]).

In the Horava–Witten solution [143 ], gauge fields of the standard model are confined on two 1+9-branes located at the end points of an $S1∕Z2$ 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 [215 , 216 , 217 ].

Figure 2:

The RS 2-brane model. (Figure taken from [58 ].)

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 [266 ] (see Figure 2 ). The single-brane 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) brane-world 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 $Mew$ is the electroweak scale. If $M4+d$ is close to the lower limit in Equation (7

), then this would address the long-standing “hierarchy” problem, i.e., why there is such a large gap between $Mew$ and $Mp$ .

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 RS2 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 RS2 model with its $AdS5$ 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 brane-worlds (mainly RS 1-brane) 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 string-theory 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 brane-worlds, I discuss the notion of Kaluza–Klein (KK) modes of the graviton.