Is four-dimensional $N=8$ supergravity ultraviolet finite to all loops?
In plain words
The most symmetric extension of Einstein gravity, $N=8$ supergravity, has unexpectedly mild short-distance behavior in explicit calculations. If it has no divergences at any order, it would be a finite perturbative quantum theory of gravity in four dimensions.
Precise statement
Compute the UV behavior of the four-graviton amplitude of $N=8$ supergravity at $L$ loops. At five loops the first divergence appears in critical dimension $D_{c}=24/5>4$ (Bern et al. 2018). Determine whether a divergence appears in $D=4$ at seven loops, where a counterterm compatible with the $E7(7)$ symmetry first exists, or at any higher loop; an answer is the coefficient of the first potential $D=4$ divergence or an all-loop finiteness proof.
What would settle it
An explicit seven-loop $D = 4$ computation of the four-graviton divergence, or a symmetry-based all-loop finiteness proof.
Status in the literature
Unverified note
The five-loop computation giving $D_c = 24/5$ (Bern et al., 2018) is the most recent explicit result; no $D = 4$ divergence has been computed as of 2026.
Related problems
- Special case of Which theory completes general relativity at the Planck scale?
- More general than Is four-dimensional $N=8$ supergravity ultraviolet finite at seven loops