sacrificing naturalness.

Our results are subject to an important subtlety that was noticed already in early studies of quantum fields in curved space: quantum inequivalence [@Duff:1980qv]. In our context an important example is the relation between a massless antisymmetric tensor and a scalar field. In the classical theory they are equivalent by a field redefinition but their quantum partition functions are related by a shift that is proportional to the Euler characteristic (\[eq:gaussbonnet\]) [@Bastianelli:2005vk; @Bastianelli:2005uy]. The coefficient of the logarithmic divergence we study therefore depends on the duality frame which becomes part of the data that defines the theory. We interpret this feature as a genuine physical effect: antisymmetric tensor fields support boundary modes that have no analogues in the corresponding scalar field theory.

In this paper we primarily interpret ${\cal N}=8$ AdS$_4$ supergravity as a low energy effective field theory in its own right but ultimately the UV completion involves the full string/M-theory. As an intermediate step we consider the theory as compactification of 11D supergravity on AdS$_4\times S^7$. This procedure defines a preferred duality frame for the massless fields and it happens that it is precisely the frame where all logarithmic divergences cancel [@Duff:1980qv]. In this setting boundary 