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 (, chapter 13)
In the models by Kaluza  and Klein  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
This can be generalized to the case of extra-dimensions  to 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  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  and Wu and Wang  in which the structure constants at lower energy are obtained by the renormalization group, and the work by Veneziano  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.  used the variation (173) to constrain the time variation of the radius of the extra dimensions during primordial nucleosynthesis to conclude that.  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 , while  assumes that gauge fields and matter fields can propagate in the bulk, that is in the extra dimensions. Ref.  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  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., ). One of the definitive predictions of these theories is the existence of a scalar field, the dilaton, that couples directly to matter  and whose vacuum expectation value determines the string coupling constant . 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. )).
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,. 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 
As an example,  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, is stabilized and the variation of the constants arises mainly from the variation of in a runaway potential.
Ref.  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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