argue that these fields are equivalent, at least for all practical purposes [@Grisaru:1984vk; @Fradkin:1984ai; @Buchbinder:2008jf].

Our discussion of divergences in ${\cal N}=8$ supergravity (and related theories) is intertwined with quantum inequivalence. First of all, the divergence (\[eq:N=8div\]) is entirely an $a$-anomaly, the $c$-anomaly of ${\cal N}=8$ supergravity vanishes. We nevertheless interpret this divergence physically in terms of the logarithmic evolution of the cosmological constant. This assigns physical significance to the $a$-anomaly even though it has a topological aspect.

Next, the value of the $a$-anomaly, and therefore its physical significance, depends on the duality frame. Concretely, one might choose to dualize any number of antisymmetric tensors into scalars, or vice versa, affecting the trace of the energy momentum tensor in the process. Therefore such dualizations are not symmetries.

AdS$_4$ SUGRA from 11D.
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The default spectrum of ${\cal N}=8$ supergravity summarized in table \[table:conformaldimensions\] comprises $70$ scalars and no antisymmetric tensors. Comparing tables \[table:centralcharges8\] and \[table:centralchargesAs\] we find that a duality frame where exactly five scalars are represented instead as antisymmetric tensors exhibits no trace anomaly.

It turns out that this precise number is a natural expectation when approaching supergravity in AdS$_4$ as compactification of 11D supergravity on $S^7$. The 11D 