  ${\cal N} =4$ super Yang-Mills theory admits \cite{Beem:2013sza} a protected
subsector isomorphic to a two-dimensional chiral algebra, obtained by passing
to the cohomology of a certain supercharge. In the large $N$ limit, we expect
this chiral algebra to have a dual description as a subsector of IIB
supergravity on $AdS_5 \times S^5$. This subsector can be carved out by a
version of supersymmetric localization, using the bulk analog of the boundary
supercharge. We illustrate this procedure in a simple model, the theory of an
${\cal N}=4$ vector multiplet in $AdS_5$, for which a convenient off-shell
description is available. This model can be viewed as the simplest truncation
of the full $AdS_5 \times S^5$ supergravity, in which case the vector multiplet
should be taken in the adjoint representation of ${\mathfrak g}_F = \mathfrak
{su}(2)_F$. Localization yields Chern-Simons theory on $AdS_3$ with gauge
algebra ${\mathfrak g}_F$, whose boundary dual is the affine Lie algebra
$\widehat {\mathfrak g}_F$. We comment on the generalization to the full bulk
theory. We propose that the large $N$ limit of the chiral algebra of ${\cal
N}=4$ SYM is again dual to Chern-Simons theory, with gauge algebra a suitable
higher-spin superalgebra.
