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N=8 SUGRA finiteness at 7 loops: 2 steps closer

In an interesting new technical preprint, an MIT-IAS-Michigan-German team proves that the maximally supersymmetric supergravity in d=4 (the low-energy limit of M-theory on a seven-torus) is free of almost all conceivable types of divergences up to the 7-loop order:

E_{7(7)} constraints on counterterms in N=8 supergravity (by Beisert, Elvang, Freedman, Kiermaier, Morales, and Stieberger)
The perturbative supergravity is known to have the exceptional E_{7(7)} symmetry classically. Because it's non-anomalous, it may be and should be preserved quantum mechanically, too - to all orders in perturbation theory. Non-perturbatively, consistency (e.g. the Dirac quantization rule for the electric and magnetic charges under U(1)^56) demands this exceptional symmetry to be broken down to its discrete E_{7(7)}(Z) U-duality subgroup, just like in M-theory on a seven-torus.

Before we impose the E_{7(7)} continuous symmetry, there are only three candidate counterterms - a counterterm is something that has to exist for every kind of a divergence. The simplest one is an R^4 term. Just two months ago, Elvang and Kiermaier proved that this term is incompatible with the continuous exceptional symmetry.

Today, the broader author team adds the remaining two dangerous candidate terms to this list, namely a D^4 R^4 term and a D^6 R^4 term - and the tower of similar operators where the superderivatives D are replaced by products of the SUGRA scalar fields. Their conclusion is that these terms violate the continuous exceptional symmetry, too. Because this partial result proves that there can't be any counterterms (respecting all the desired symmetries, including SUSY and the continuous exceptional Lie group) up to the seven-loop level, there can be no divergences, either.

Henriette has contacted me to point out that there is also a D^8 R^4 term that they haven't eliminated. So as a proof of finiteness, the paper remains incomplete. For the rest of my text, let me assume that this candidate counterterm violates the symmetry, too. (Even if it doesn't violate it, the theory may stay finite at 7 loops if it manages to produce a vanishing coefficient of this term.) That would mean that the theory is 7-loop finite.

Although seven looks like a very high number, it is actually not "quite" excluded that there could be divergences at even higher loops. In particular, new things could take place at the 8-loop and 9-loop level. However, the tantalizing similarity to the N=4 gauge theory (squared) which is UV finite indicate that the perturbative finiteness will survive to all loops.

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