E{7(7)} Symmetry and Finiteness of N=8 Supergravity

TL;DR

The paper probes whether ${\rm N}=8$ supergravity remains UV finite when incorporating candidate ${E_{7(7)}}$-invariant counterterms. By applying the NGZ Noether-Gaillard-Zumino duality framework and exploiting the Cremmer-Julia constraint that reduces the 56-vector system to 28 physical vectors, it shows that deformations induced by counterterms violate ${E_{7(7)}}$ current conservation or conflict with the unique vector-duality relation. A concrete 3-loop sector analysis confirms that the NGZ identity cannot be satisfied unless counterterms vanish, and a unitarity-based argument reinforces this conclusion via the rigidity of the ${SU(8)}$–to–${E_{7(7)}}$ mapping. Together, these results argue that all gauge-symmetric, ${E_{7(7)}}$-invariant counterterms are forbidden, supporting all-loop perturbative finiteness of ${\rm N}=8$ supergravity in the absence of anomalies.

Abstract

We study N=8 supergravity deformed by the presence of the candidate counterterms. We show that even though they are invariant under undeformed E{7(7)}, all of the candidate counterterms violate the deformed E{7(7)} current conservation. The same conclusion follows from the uniqueness of the Lorentz and SU(8) covariant, E{7(7)} invariant unitarity constraint expressing the 56-dimensional E{7(7)} doublet via 28 independent vectors. Therefore E{7(7)} duality predicts the all-loop UV finiteness of perturbative N=8 supergravity.