from table \[table:centralcharges\] we find that the central charge $c=0$ for the full ${\cal N}=8$ multiplet. We collect these results in table \[table:centralcharges8\].

The quantum effective action can be computed from the heat kernel (or, equivalently, from the trace of the energy momentum tensor) by the integral (\[eq:Wdef\]). We perform the integration with the dimensionless conformal weights $\Delta$ kept fixed. This is justified by the boundary perspective where the dual theory is conformal in the leading approximation and also from the bulk point of view where all fields are in the massless representations that do not even exist for other values of the conformal weights. Since we focus on theories with no power law corrections the integrand is a constant and, with the measure indicated in (\[eq:Wdef\]), the integral gives a logarithmically divergent term in the effective action.

Multiple research groups have reported that in fact the trace anomaly does vanish for ${\cal N}=8$ supergravity in AdS$_4$ [@Gibbons:1984dg; @Inami:1984vp] and so there are no divergences. Those results refer to different boundary conditions where the spectrum is discrete and the helicity sum rule (\[eq:supertrace\]) applies for all $n<{\cal N}$. We will return to this in more detail in the next section.

                                        