Supersymmetric Localization in AdS${}_5$ and the Protected Chiral Algebra

$\mathcal{N}=4$ 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 $Ad{S}_5\times S^5$. 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 $\mathcal{N}=4$ vector multiplet in $Ad{S}_5$, for which a convenient off-shell description is available. This model can be viewed as the simplest truncation of the full $Ad{S}_5\times S^5$ supergravity, in which case the vector multiplet should be taken in the adjoint representation of ${\mathfrak g}_F = \mathfrak {su}(2)_F$\mathrm{.}LocalizationyieldsChern-Simonstheoryon$AdS_3$ 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.