Such coupling terms naturally appear when compactifying a higher-dimensional theory. As an example, let us recall the compactification of a 5-dimensional Einstein–Hilbert action ([409], chapter 13)

In the models by Kaluza [269] and Klein [291] the 5-dimensional spacetime was compactified assuming that one spatial extra-dimension , of radius . It follows that any field ) can be Fourier transformed along the compact dimension (with coordinate ), so that, from a 4-dimensional point of view, it gives rise to a tower of of fields of mas . At energies small compared to only the -independent part of the field remains and the physics looks 4-dimensional.

Assuming that the action (171) corresponds to the Jordan frame action, as the coupling may suggest, it follows that the gravitational constant and the Yang–Mills coupling associated with the vector field must scale as

Note that the scaling of with (or time) is not the one of the gravitational constant that would be measured in a Cavendish experiment since Eq. (165) tells us that .This can be generalized to the case of extra-dimensions [114] to

where the constants depends only on the dimension and topology of the compact space [525] so that the only fundamental constant of the theory is the mass scale entering the -dimensional theory. A theory on where is a -dimensional compact space generates a low-energy quantum field theory of the Yang–Mills type related to the isometries of [for instance [545] showed that for , it can accommodate the Yang–Mills group ]. The two main problems of these theories are that one cannot construct chiral fermions in four dimensions by compactification on a smooth manifold with such a procedure and that gauge theories in five dimensions or more are not renormalizable.In such a framework the variation of the gauge couplings and of the gravitational constant arises from the variation of the size of the extra dimensions so that one can derives stronger constraints that by assuming independent variation, but at the expense of being more model-dependent. Let us mention the works by Marciano [345] and Wu and Wang [550] in which the structure constants at lower energy are obtained by the renormalization group, and the work by Veneziano [515] for a toy model in dimensions, endowed with an invariant UV cut-off , and containing a large number of non-self-interacting matter species.

Ref. [295] used the variation (173) to constrain the time variation of the radius of the extra dimensions
during primordial nucleosynthesis to conclude that. [28] took the effects of the
variation of and deduced from the helium-4 abundance that
and respectively for and Kaluza–Klein theory and
that from the Oklo data. An analysis of most cosmological data
(BBN, CMB, quasar etc..) assuming that the extra dimension scales as and
concluded that has to be smaller than 10^{–16} and 10^{–8} respectively [311],
while [330] assumes that gauge fields and matter fields can propagate in the bulk, that is in the extra
dimensions. Ref. [336] evaluated the effect of such a couple variation of and the structures constants on
distant supernova data, concluding that a variation similar to the one reported in [524] would make the
distant supernovae brighter.

There exist five anomaly-free, supersymmetric perturbative string theories respectively known as type I, type IIA, type IIB, SO(32) heterotic and heterotic theories (see, e.g., [420]). One of the definitive predictions of these theories is the existence of a scalar field, the dilaton, that couples directly to matter [484] and whose vacuum expectation value determines the string coupling constant [546]. There are two other excitations that are common to all perturbative string theories, a rank two symmetric tensor (the graviton) and a rank two antisymmetric tensor . The field content then differs from one theory to another. It follows that the 4-dimensional couplings are determined in terms of a string scale and various dynamical fields (dilaton, volume of compact space, …). When the dilaton is massless, we expect three effects: (i) a scalar admixture of a scalar component inducing deviations from general relativity in gravitational effects, (ii) a variation of the couplings and (iii) a violation of the weak equivalence principle. Our purpose is to show how the 4-dimensional couplings are related to the string mass scale, to the dilaton and the structure of the extra dimensions mainly on the example of heterotic theories.

To be more specific, let us consider an example. The two heterotic theories originate from the fact that left- and right-moving modes of a closed string are independent. This reduces the number of supersymmetry to and the quantization of the left-moving modes imposes that the gauge group is either or depending on the fermionic boundary conditions. The effective tree-level action is (see, e.g., Ref. [237])

When compactified on a 6-dimensional Calabi–Yau space, the effective 4-dimensional action takes the form where couples identically to the Einstein and Yang–Mills terms. It follows that at tree-level. Note that to reach this conclusion, one has to assume that the matter fields (in the ‘dots’ of Equation (175) are minimally coupled to ; see, e.g., [340]).The strongly coupled heterotic string theory is equivalent to the weakly coupled type I string theory. Type I superstring admits open strings, the boundary conditions of which divide the number of supersymmetries by two. It follows that the tree-level effective bosonic action is , supergravity, which takes the form, in the string frame,

where the dots contains terms describing the dynamics of the dilaton, fermions and other form fields. At variance with (174), the field couples differently to the gravitational and Yang–Mills terms because the graviton and Yang–Mills fields are respectively excitation of close and open strings. It follows that can be lowered even to the weak scale by simply having small enough. Type I theories require -branes for consistency. When is small, one can use T-duality (to render large, which allows to use a quantum field theory approach) and turn the -brane into a -brane so that where the second term describes the Yang–Mills fields localized on the -brane. It follows that at tree-level. If one compactifies the -brane on a 6-dimensional orbifold instead of a 6-torus, and if the brane is localized at an orbifold fixed point, then gauge fields couple to fields living only at these orbifold fixed points with a (calculable) tree-level coupling so that The coupling to the field is a priori non universal. At strong coupling, the 10-dimensional heterotic theory becomes M-theory on [255]. The gravitational field propagates in the 11-dimensional space while the gauge fields are localized on two 10-dimensional branes.At one-loop, one can derive the couplings by including Kaluza–Klein excitations to get [163]

when the volume is large compared to the mass scale and in that case the coupling is no more universal. Otherwise, one would get a more complicated function. Obviously, the 4-dimensional effective gravitational and Yang–Mills couplings depend on the considered superstring theory, on the compactification scheme but in any case they depend on the dilaton.As an example, [340] considered the ()-supergravity model derived from the heterotic superstring theory in the low energy limit and assumed that the 10-dimensional spacetime is compactified on a 6-torus of radius so that the effective 4-dimensional theory described by (175) is of the Brans–Dicke type with . Assuming that has a mass , and couples to the matter fluid in the universe as , the reduced 4-dimensional matter action is

The cosmological evolution of and can then be computed to deduce that . considered the same model but assumed that supersymmetry is broken by non-perturbative effects such as gaugino condensation. In this model, and contrary to [340], is stabilized and the variation of the constants arises mainly from the variation of in a runaway potential.Ref. [290] considers a probe D3-brane probe in the context of AdS/CFT correspondence at finite temperature and provides the predictions for the running electric and magnetic effective couplings, beyond perturbation theory. It allows to construct a varying speed of light model.

To conclude, superstring theories offer a natural theoretical framework to discuss the value of the fundamental constants since they become expectation values of some fields. This is a first step towards their understanding but yet, no complete and satisfactory mechanism for the stabilization of the extra dimensions and dilaton is known.

It has paved the way for various models that we detail in Section 5.4.

Living Rev. Relativity 14, (2011), 2
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