Is four-dimensional $N=8$ supergravity ultraviolet finite at seven loops
In plain words
$N=8$ supergravity is the most symmetric theory of gravity in four dimensions and might lack the infinities that ruin ordinary quantum gravity. Known symmetries first allow an infinity at seven loops, a calculation not yet done.
Precise statement
Compute the coefficient of the seven-loop logarithmic ultraviolet divergence (the $D^{8}R^{4}$ counterterm) of the four-graviton amplitude in $D=4$ $N=8$ supergravity, or prove it vanishes. The highest computed order gives critical dimension $D_{c}=24/5$ at five loops (Bern et al., Phys. Rev. D 98, 086021, 2018).
What would settle it
Construction of the seven-loop four-graviton integrand and extraction of its $D = 4$ ultraviolet divergence, or a symmetry argument forbidding the counterterm.
Status in the literature
Unverified note
No six- or seven-loop result reported as of 2026.
Related problems
- Special case of Is four-dimensional $N=8$ supergravity ultraviolet finite to all loops? · Seven-loop finiteness is the first critical case of all-loop finiteness