modes cancel divergences.

There have been many other recent studies of quantum corrections to AdS spaces in various dimensions. A basic feature of this research is that divergences remain even when supersymmetry is maximal and those divergences are related to effects that are unambiguously physical in the dual theory. Some examples:

-   In AdS$_{d+1}$ with odd $(d+1)$ there are bulk divergences interpreted as finite quantum anomalies in the dual theory with even $d$. For example, in the case of $d=4$ such anomalies are responsible for the shift $N^2\to N^2-1$ that is expected and confirmed in ${\cal N}=4$ SYM with $SU(N)$ gauge groups [@Mansfield:2003gs; @Beccaria:2014xda].

-   Quantum corrections to higher spin theories in AdS provide impressive evidence for higher spin holography. [@Giombi:2013fka; @Giombi:2014iua; @Giombi:2014yra]

-   The Bekenstein-Hawking area law for black holes is subject to $\log A$ corrections with coefficients determined by the low energy theory. For BPS black holes these coefficients are determined by divergences in AdS$_2$ and AdS$_2\times S^2$ which are generically nonvanishing (including for ${\cal N}=8$), and their values are confirmed by the microscopic theory in cases where the latter has been established. [@Banerjee:2010qc; @Banerjee:2011jp; @Sen:2011ba; @Sen:2012cj; @Bhattacharyya:2012wz; @Sen:2012dw; @Sen:2014aja; @Keeler:2014bra; @Larsen:2014bqa]

Our study of AdS$_4$ was motivated 