The Ultraviolet Finiteness of N=8 Supergravity

TL;DR

The paper investigates the ultraviolet behavior of four-dimensional $ ext{N}=8$ supergravity by constructing and cross-examining counterterms in Lorentz-covariant on-shell, real LC, and chiral LC superspaces. It demonstrates that every candidate CT either fails linearized or nonlinear Ward identities or cannot be realized as a local D-term in real LC superspace, revealing a vanishing intersection of the three CT-sets. This leads to a perturbative UV finiteness claim, contingent on anomaly-free $ ext{E}_{7(7)}$ symmetry; the equivalence theorem tying the three formalisms under anomaly freedom is central. The analysis leverages F-term vs D-term distinctions in LC superspace and the structure of $ ext{N}^k ext{MHV}$ amplitudes, and it highlights a key difference from string theory due to the absence of 1/2-BPS states in perturbative loops. Overall, the work supports all-loop finiteness of $ ext{N}=8$ supergravity in 4D, pending the absence of perturbative anomalies in $ ext{E}_{7(7)}$.

Abstract

We study counterterms (CT's), candidates for UV divergences in the four-dimensional N=8 supergravity. They have been constructed long ago in a Lorentz covariant on shell superspace and recently in the chiral light-cone (LC) superspace. We prove that all of these CT's are ruled out since they are not available in the real LC superspace. This implies the perturbative UV finiteness of d=4 N=8 supergravity under the assumption that supersymmetry and continuous E7 symmetry are anomaly-free. The proof, based on the chiral nature of CT's in the LC superspace, is a generalization of the perturbative F-term non-renormalization theorem for N=8 supergravity.