This paper seems to show that $d=4, N=8$ supergravity is finite. Yet the paper only has three citations in spires, and I certainly haven't heard talk of a new candidate theory of gravity.
Why isn't perturbative supergravity with some supersymmetry breaking principle, coupled with the standard model considered a possible theory of the universe? Has someone checked the coupling to matter? Is that the problem?
In order to not be entirely negative, I'll answer your question first and then provide a reason or two why research on the subject is interesting for other reasons.
The finiteness conjecture has to do with perturbation theory. Even if true, it is still believed that in order for the theory to be non-perturbatively finite and otherwise consistent (e.g unitary), it has to include more degrees of freedom. The most plausible scenario is that the completion is the full string theory, for which N=8 SUGRA is very closely related.
The theory does not have enough structure to be a realistic description of nature by itself. For example it does not have chiral fermions, or a mechanism to break the extended SUSY spontaneously. All of these things become possible when you embed the theory within string theory.
If you keep the theory as a QFT, but add some ingredients by hand to get more a realistic theory, the extended SUSY is broken and the magic is gone. Any finiteness conjecture is only valid for the pure (super)gravity case, and goes away as soon as you make the situation slightly less symmetric.
On the other hand, purely as a theoretical laboratory it is fascinating that a quantum field theory which contains gravity is better behaved than expected, and the theory has a close relationship to other highly symmetric quantum field theories (for example N=4 SYM). There is great excitement and hope recently that by understanding precisely why it is finite (or at least well-behaved) we'd be able to understand the structure of quantum field theories better. It is in some sense the "simplest quantum field theory" (arxiv.org/abs/0808.1446).
Dear Jerry, the $N=8$, $d=4$ "non-stringy" supergravity is
Trying to fix either of these things leads one to string/M-theory. See
Two roads from $N=8$ sugra to string theory http://motls.blogspot.com/2008/07/two-roads-from-n8-sugra-to-string.html
The perturbative inconsistency may be seen in many ways: for example, the supergravity theory has $U(1)$ charges but produces no charged objects with respect to these $U(1)$'s. That's inconsistent because at least a newly formed black hole may confine these electric and magnetic fields and become charged.
The electric and magnetic charges have to be quantized in inverse units, as seen by the Dirac quantization argument. It follows that the noncompact continuous exceptional $E_{7(7)}$ symmetry has to be broken to its discrete subgroup, the U-duality group. There are many ways to choose the lattice of allowed charges. These ways are related by the original continuous symmetry. In decompactification limits, the lightest of these charges (with smallest spacing) may be interpreted as Kaluza-Klein momenta with respect to new dimensions, and one discovers the 7 compactified dimensions of M-theory. It may also be showed that the other charges inevitably have the shape of string/M-theoretical membranes and fivebranes.
There's no doubt today - and since the mid 1990s, in fact - that the supergravity theory is just a perturbative approximation to string/M-theory which is also why the supergravity community has been fully merged with the string/M-theory community. The people realize that they are working on the same theory and they are saying the same things. Ask Michael Duff.
Phenomenology
The maximal supergravity in four dimensions is left-right symmetric, and the high supersymmetry leads to too huge degenerate multiplets where spins differ by as much as $2$. The only acceptable supersymmetry is the minimal one where spins differ by $1/2$. The maximum supersymmetry implies that left-handed neutrinos couldn't exist and for each particle, there would have to be lots of very different superpartners. One couldn't get matter and gauge fields decoupled from gravity etc.
The maximum supersymmetry cannot be broken down to a smaller one by field-theoretical mechanisms - except for an explicit breaking that just destroys all the finiteness virtues of the supergravity. However, it may be broken at the stringy level, by appreciating the extra 6-7 dimensions, and compactifying them differently. The resulting models are compactifications of string/M-theory. They preserve the perturbative finiteness by the added stringy species and they also lead to realistic phenomenology with all types of matter and interactions that we know.
As Joe Polchinski said, all roads lead to string theory. In the case of attempts to overcome limitations of supergravity, the previous sentence is not a slogan but rather an accurate description of the situation.
Cheers LM
There has been a recent spate of interest in computing high loop quantum gravity without strings. In my opinion this is a huge laborious effort to capture what lies in string theory already. $N = 8$ SUGRA with an $SU(8)$ symmetry plus an $E_{7(7)}$ symmetry which acts independently. The $SU(8)$ acts upon the $133$ dimensions of the $E_(7(7)$, which we can think of for $E_{7(7)}({\bf R})$ as $133$ scalars. The $133$ real parameters of $E_{7(7)}$ are ${\bf 133}~=~{\bf 28}~+~{\bf 35}~+~{\bf 35}~+~{\bf 35}$, where $\bf 28$ is an $SO(8)$ and combined with ${\bf 35}$ is an $SU(8)$. The current is composed of these scalars as $$ J_\mu=~\sum_{a=1}^{133}J_\mu^a e_a $$ with the current conservation rule $\partial^\mu J_\mu~=~0$ If with fix the $SU(8)$ this is a coset rule $E_{7(7)}/SU(8)$ which subtracts out the ${\bf 28}~+~{\bf 35}~=~{\bf 63}$ of $SU(8)$ are leaves $70$ scalars. Thus we may think of $E_{7(7)}$ as $$ E_{7(7)}~=~SU(8)\times R^{70}. $$ The coset construction removes trouble some terms or provide counter terms. It is then possible to compute currents, which are Noetherian conserved, in this coset construction which up to $7$ loops is $UV$ finite.
The $E_{7(7)}({\bf R})$ is broken into a discrete group $ E_{7(7)}({\bf Z})$ by “quantization,” which in turn contains the modular (or Mobius) group $SL(2,{\bf Z})$ of S duality and the T-duality group $SO(6,6,{\bf Z})$. The Noether theorem operates for continuous symmetries, not for discrete ones. However, a braid group or Yang-Baxter arguments may recover current conservation. Further, a more general U duality description which meshes S an T dualities together may conserve current. So there are open questions with the construction which have not been answered. In an STU setting the lack of Noetherian currents should be replaced with Noetherian charges associated with qubits.
For $N~=~8$ and $d~=~4$ a paper by Green, Ooguri, Schwarz
http://arxiv.org/PS_cache/arxiv/pdf/0704/0704.0777v1.pdf
it is not possible to decouple string theory from SUGRA. There exists a set of states which make any decoupling inconsistent. This calls into question the ability to compute the appropriate counter terms required to make a consistent SUGRA which is $UV$ finite. This paper attempts to circumvent this problem, but it must be realized this is a $7$ loop computation of considerable complexity which recovers results of string theory that are obtained rather readily.
These efforts are not without purpose though, for the construction of Noetherian currents, which I think will correspond to Noetherian charges associated with qubits and N-partite entanglements in STU theories. However, as a replacement for string theory, reduction of gravity to a pure QFT, I doubt this will ever prove to be a complete approach.
