The observation is that the perturbative spectrum of $\mathcal{N}$=8 supergravity in four dimensions is exactly the same as the type IIB closed superstring spectrum after dimensional reduction on $T^{6}$ wich is in turn equivalent to M-theory compactified on $T^{7}$ (after a T-duality on an internal direction is performed). The latter identitication is useful because it gives an straightforward way to construct and compute the dimension of the the scalar and vector moduli spaces as 28 scalars of the form $G_{nm}$ and other 35 scalars descendin from the 3-form as $A_{nmp}$ where n,m are tangential indices on $T^{7}$ and the p is transversal to it. So globally the scalar moduli space is $$E_{7(7)}(\mathbb{Z})\diagdown \ E_{7(7)}/ SU(8)$$
here $E_{7(7)}$ is the noncompact version of $E_{7}$ and $E_{7(7)}(\mathbb{Z})$ is the U duality group of the type IIB string compactified on $T^{6}$. For details see section 3.2 on the page 23 of the document Introduction to M-theory. This argument seems to intuitively explain the "origin" of the hidden $E_{7(7)}$ symmetry in ($\mathcal{N}=8 , $d$=4$) supergravity.
Another striking coincidence is the fact that the maximal supergravity in four dimensions does not admit perturbative non-abelian deformations, that seems remminiscent from the fact that there is no possibility of non-abelian symmetries or gauge symmetry enhancement in the case of the IIb theory on $T^{6}$
Nevertheles the maximal supergravity in four dimensions can't be the effective theory arising from type IIB theory on $T^{6}$ because it was demostrated (here) that certain massless and finite mass states of the string theory cannot be decoupled from the perturbative spectrum in the zero slope limit.
Even something naive such as taking something like the Vol($T^{6}$) $\rightarrow 0$ limit of type IIB string compactified on a six-torus to try to avoid the extra stringy states present in the supergravity approximation to the latter theory but abscent in the $\mathcal{N}=8$ , $d$=4 theory, then we are in trouble because now extra Kahler moduli (associated to the $T^{6}$ volume) become light and enter into the perturbative spectrum. Not to mention the need of a moduli stabilization mechanism.
Question: If the maximal supergravity in four dimensions cannot be a the compactification of the type IIB theory on $T^{6}$ why the two theories share the same spectrum? Is there any explanation/speculation of about why this is true?
Thanks in advance :)
